[Pas] Probability Abstracts 95
pas at www2.economia.unimi.it
pas at www2.economia.unimi.it
Mon Jan 1 15:25:59 CET 2007
Jan 1st, 2007
Letter 95
Probability Abstracts Service
This document contains abstracts 4722-5092 from
Oct-1-2006 to Dic-31-2006.
They have been mailed on Jan 1st, 2007.
This letter can be also found on line at
http://www2.economia.unimi.it/PAS/Letters/letter_95.shtml
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4722. CONDENSATION FOR A FIXED NUMBER OF INDEPENDENT RANDOM VARIABLES
Pablo A. Ferrari and Claudio Landim and Valentin V. Sisko
A family of m independent identically distributed random variables
indexed by
a chemical potential \phi\in[0,\gamma] represents piles of particles.
As \phi
increases to \gamma, the mean number of particles per site converges
to a
maximal density \rho_c<\infty. The distribution of particles
conditioned on the
total number of particles equal to n does not depend on \phi (canonical
ensemble). For fixed m, as n goes to infinity the canonical ensemble
measure
behave as follows: removing the site with the maximal number of
particles, the
distribution of particles in the remaining sites converges to the grand
canonical measure with density \rho_c; the remaining particles
concentrate
(condensate) on a single site.
http://front.math.ucdavis.edu/math.PR/0612856
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4723. SURVIVAL PROBABILITY OF A DIFFUSING PARTICLE CONSTRAINED BY TWO
MOVING, ABSORBING BOUNDARIES
Alan J. Bray and Richard Smith
We calculate the exact asymptotic survival probability, Q, of a
one-dimensional Brownian particle, initially located located at the
point x in
(-L,L), in the presence of two moving absorbing boundaries located at
\pm(L+ct). The result is Q(y,\lambda) = \sum_{n=-\infty}^\infty \cosh
(ny)
\exp(-n^2\lambda/4), where y=cx/D, \lambda = cL/D and D is the diffusion
constant of the particle. The results may be extended to the case
where the
absorbing boundaries have different speeds. As an application, we
compute the
asymptotic survival probability for the trapping reaction A + B -> B,
for
evanescent traps with a long decay time.
http://front.math.ucdavis.edu/cond-mat/0612563
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4724. HIGHLY ROBUST ERROR CORRECTION BY CONVEX PROGRAMMING
Emmanuel J. Candes and Paige A. Randall
This paper discusses a stylized communications problem where one
wishes to
transmit a real-valued signal x in R^n (a block of n pieces of
information) to
a remote receiver. We ask whether it is possible to transmit this
information
reliably when a fraction of the transmitted codeword is corrupted by
arbitrary
gross errors, and when in addition, all the entries of the codeword are
contaminated by smaller errors (e.g. quantization errors).
We show that if one encodes the information as Ax where A is a
suitable m by
n coding matrix (m >= n), there are two decoding schemes that allow the
recovery of the block of n pieces of information x with nearly the same
accuracy as if no gross errors occur upon transmission (or
equivalently as if
one has an oracle supplying perfect information about the sites and
amplitudes
of the gross errors). Moreover, both decoding strategies are very
concrete and
only involve solving simple convex optimization programs, either a
linear
program or a second-order cone program. We complement our study with
numerical
simulations showing that the encoder/decoder pair performs remarkably
well.
http://front.math.ucdavis.edu/cs.IT/0612124
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4725. BILLIARDS IN A GENERAL DOMAIN WITH RANDOM REFLECTIONS
Francis Comets and Serguei Popov and Gunter Sch\"utz and Marina
Vachkovskaia
We study stochastic billiards on general tables: a particle moves
according
to its constant velocity inside some domain ${\mathcal D} \subset
{\mathbb
R}^d$ until it hits the boundary and bounces randomly inside
according to some
reflection law. We assume that the boundary of the domain is locally
Lipschitz
and almost everywhere continuously differentiable. The angle of the
outgoing
velocity with the inner normal vector has a specified, absolutely
continuous
density. We construct the discrete time and the continuous time
processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains.
Then, we
prove exponential ergodicity of these two Markov processes, we study
their
invariant distribution and their normal (Gaussian) fluctuations. Of
particular
interest is the case of the cosine reflection law: the stationary
distributions
for the two processes are uniform in this case, the discrete time
chain is
reversible though the continuous time process is quasi-reversible.
Also in this
case, we give a natural construction of a chord ``picked at random'' in
${\mathcal D}$, and we study the angle of intersection of the process
with a
$(d-1)$-dimensional manifold contained in ${\mathcal D}$.
http://front.math.ucdavis.edu/math.PR/0612799
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4726. UNIQUENESS AND NON-UNIQUENESS IN PERCOLATION THEORY
Olle H\"{a}ggstr\"{o}m and Johan Jonasson
This paper is an up-to-date introduction to the problem of uniqueness
versus
non-uniqueness of infinite clusters for percolation on ${\mathbb{Z}}^d
$ and,
more generally, on transitive graphs. For iid percolation on ${\mathbb
{Z}}^d$,
uniqueness of the infinite cluster is a classical result, while on
certain
other transitive graphs uniqueness may fail. Key properties of the
graphs in
this context turn out to be amenability and nonamenability. The same
problem is
considered for certain dependent percolation models -- most
prominently the
Fortuin--Kasteleyn random-cluster model -- and in situations where
the standard
connectivity notion is replaced by entanglement or rigidity. So-called
simultaneous uniqueness in couplings of percolation processes is also
considered. Some of the main results are proved in detail, while for
others the
proofs are merely sketched, and for yet others they are omitted.
Several open
problems are discussed.
http://front.math.ucdavis.edu/math.PR/0612812
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4727. ASYMPTOTIC NORMALITY OF THE K-CORE IN RANDOM GRAPHS
Svante Janson and Malwina J. Luczak
We study the $k$-core of a random (multi)graph on $n$ vertices with a
given
degree sequence. In our previous paper `A simple solution to the k-core
problem' we used properties of empirical distributions of independent
random
variables to give a simple proof of the fact that the size of the giant
$k$-core obeys a law of large numbers as $n$ tends to infinity. Here
we develop
the method further and show that the fluctuations around the
deterministic
limit converge to a Gaussian law above and near the threshold, and to a
non-normal law at the threshold. Further, we determine precisely the
location
of the phase transition window for the emergence of a giant $k$-core.
Hence we
deduce corresponding results for the $k$-core in $G(n,p)$ and $G(n,m)$.
http://front.math.ucdavis.edu/math.PR/0612827
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4728. LARGE DEVIATIONS AND RANDOM ENERGY MODELS
N. K. Jana and B. V. Rao
A unified treatment for the existence of free energy in several
random energy
models is presented. If the sequence of distributions associated with
the
particle systems obeys a large deviation principle, then the free
energy exists
almost surely. This includes all the known cases as well as some
heavy-tailed
distributions.
http://front.math.ucdavis.edu/math.PR/0612836
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4729. HIGH DIMENSIONAL PROBABILITY
Evarist Gin\'{e} and Vladimir Koltchinskii and Wenbo Li and Joel Zinn
About forty years ago it was realized by several researchers that the
essential features of certain objects of Probability theory, notably
Gaussian
processes and limit theorems, may be better understood if they are
considered
in settings that do not impose structures extraneous to the problems
at hand.
For instance, in the case of sample continuity and boundedness of
Gaussian
processes, the essential feature is the metric or pseudometric structure
induced on the index set by the covariance structure of the process,
regardless
of what the index set may be. This point of view ultimately led to the
Fernique-Talagrand majorizing measure characterization of sample
boundedness
and continuity of Gaussian processes, thus solving an important
problem posed
by Kolmogorov. Similarly, separable Banach spaces provided a minimal
setting
for the law of large numbers, the central limit theorem and the law
of the
iterated logarithm, and this led to the elucidation of the minimal
(necessary
and/or sufficient) geometric properties of the space under which
different
forms of these theorems hold. However, in light of renewed interest in
Empirical processes, a subject that has considerably influenced modern
Statistics, one had to deal with a non-separable Banach space, namely
$\mathcal{L}_{\infty}$. With separability discarded, the techniques
developed
for Gaussian processes and for limit theorems and inequalities in
separable
Banach spaces, together with combinatorial techniques, led to powerful
inequalities and limit theorems for sums of independent bounded
processes over
general index sets, or, in other words, for general empirical processes.
http://front.math.ucdavis.edu/math.PR/0612726
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4730. LAW OF THE ITERATED LOGARITHM FOR STATIONARY PROCESSES
Ou Zhao and Michael Woodroofe
There has been recent interest in the conditional central limit
question for
(strictly) stationary, ergodic processes $... X_{-1}, X_0,X_1,...$ whose
partial sums $S_n = X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where
$M_n$ is
a square integrable martingale with stationary increments and $R_n$ is a
remainder term for which $E(R_n^2) = o(n)$. Here we explore the Law
of the
Iterated Logarithm (LIL) for the same class of processes. Letting
$\Vert\cdot\Vert$ denote the norm in $L^2(P)$, a sufficient condition
for the
partial sums of a stationary process to have the form $S_n = M_n+R_n$
is that
$n^{-{3\over 2}}\Vert E(S_n|X_0,X_{-1},...)\Vert$ be summable. A
sufficient
condition for the LIL is only slightly stronger, requiring $n^{-{3\over
2}}\log^{3\over 2} (n)\Vert E(S_n|X_0,X_{-1},...)\Vert$ to be
summable. As a
by-product of our main result, we obtain an improved statement of the
Conditional Central Limit Theorem. Invariance principles are obtained
as well.
http://front.math.ucdavis.edu/math.PR/0612747
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4731. CONVEX CHAINS IN Z^2
Nathanael Enriquez (PMA)
A detailed combinatorial analysis of lattice convex polygonal lines
of N^2
joining 0 to (n,n) is presented. We derive consequences on the line
having the
largest number of vertices as well as the cardinal and limit shape of
lines
having few vertices. The proof refines a statistical physical method
used by
Sinai to obtain the typical behavior of these lines, allied to some
Fourier
analysis. Limit shapes of convex lines joining 0 to (n,n) and having
a given
total length are also characterized.
http://front.math.ucdavis.edu/math.PR/0612770
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4732. EMPIRICAL GRAPH LAPLACIAN APPROXIMATION OF LAPLACE--BELTRAMI
OPERATORS: LARGE SAMPLE RESULTS
Evarist Gin\'{e} and Vladimir Koltchinskii
Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of
dimension
$\scriptstyle{d}$ and let ${X_1,...,X_n}$ be a sample of i.i.d.
points in ${M}$
with uniform distribution. We study the random operators $$
\Delta_{h_n,n}f(p):=\frac{1}{nh_n^{d+2}}\sum_{i=1}^n
K(\frac{p-X_i}{h_n})(f(X_i)-f(p)), p\in M $$ where
${K(u):={\frac{1}{(4\pi)^{d/2}}}e^{-\|u\|^2/4}}$ is the Gaussian
kernel and
${h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph
laplacians (for a weighted graph with vertices at data points) and
they have
been used in the machine learning literature to approximate the
Laplace-Beltrami operator of ${M,}$ ${\Delta_Mf}$ (divided by the
Riemannian
volume of the manifold). We prove several results on a.s. and
distributional
convergence of the deviations
${\Delta_{h_n,n}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ for smooth
functions ${f}$
both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)}
$ and
${\mu}$ is the Riemannian volume measure). In particular, we show
that for any
class ${{\cal F}}$ of three times differentiable functions on ${M}$ with
uniformly bounded derivatives $$ \sup_{p\in M}\sup_{f\in
F}\Big|\Delta_{h_n,p}f(p)-\frac{1}{|\mu|}\Delta_Mf(p)\Big|=
O\Big(\sqrt{\frac{\log(1/h_n)}{nh_n^{d+2}}}\Big) a.s. $$ as soon as $$
nh_n^{d+2}/\log h_n^{-1}\to \infty and nh^{d+4}_n/\log h_n^{-1}\to 0,
$$ and
also prove asymptotic normality of
${\Delta_{h_n,p}f(p)-{\frac{1}{|\mu|}}\Delta_Mf(p)}$ (functional CLT)
for a
fixed ${p\in M}$ and uniformly in ${f}.$
http://front.math.ucdavis.edu/math.PR/0612777
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4733. ESTIMATION FOR THE DISCRETELY OBSERVED TELEGRAPH PROCESS
stefano m. iacus and nakahiro yoshida
The telegraph process $\{X(t), t>0\}$, is supposed to be observed at
$n+1$
equidistant time points $t_i=i\Delta_n,i=0,1,..., n$. The unknown
value of
$\lambda$, the underlying rate of the Poisson process, is a parameter
to be
estimated. The asymptotic framework considered is the following:
$\Delta_n \to 0$, $n\Delta_n = T \to \infty$ as $n \to \infty$. We
show that
previously proposed moment type estimators are consistent and
asymptotically
normal but not efficient. We study further an approximated moment type
estimator which is still not efficient but comes in explicit form.
For this
estimator the additional assumption $n\Delta_n^3 \to 0$ is required
in order to
obtain asymptotic normality. Finally, we propose a new estimator
which is
consistent, asymptotically normal and asymptotically efficient under no
additional hypotheses.
http://front.math.ucdavis.edu/math.PR/0612784
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4734. A CLT FOR REGULARIZED SAMPLE COVARIANCE MATRICES
Greg W Anderson and Ofer Zeitouni
We consider the spectral properties of a class of {\em regularized
estimators} of (large) empirical covariance matrices corresponding to
stationary (but not necessarily Gaussian) sequences, obtained by {\em
banding}.
We prove a law of large numbers (similar to that proved in the
Gaussian case by
Bickel and Levina), which implies that the spectrum of a banded
empirical
covariance matrix is an efficient estimator. Our main result is a
central limit
theorem in the same regime, which to our knowledge is new, even in
the Gaussian
setup.
http://front.math.ucdavis.edu/math.PR/0612791
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4735. LAW OF LARGE NUMBERS FOR SUPERDIFFUSIONS: THE NON-ERGODIC CASE
Janos Englander
In a previous paper of Winter and the author the Law of Large Numbers
for the
local mass of certain superdiffusions was proved under a spectral
theoretical
assumption, which is equivalent to the ergodicity (positive
recurrence) of the
motion component of an $H$-transformed (or weighted) superprocess. In
fact the
assumption is also equivalent to the property that the scaling for the
expectation of the local mass is pure exponential.
In this paper we go beyond ergodicity, that is we consider cases
when the
scaling is not purely exponential. Inter alia, we prove the analog of
the
Watanabe-Biggins Law of Large Numbers for super-Brownian motion (SBM).
We will also prove another Law of Large Numbers for a bounded set
moving with
subcritical speed, provided the variance term decays sufficiently fast.
http://front.math.ucdavis.edu/math.PR/0612797
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4736. STOCHASTIC INERTIAL MANIFOLDS FOR DAMPED WAVE EQUATIONS
Zhenxin Liu
In this paper, stochastic inertial manifold for damped wave equations
subjected to additive white noise is constructed by the Lyapunov-
Perron method.
It is proved that when the intensity of noise tends to zero the
stochastic
inertial manifold converges to its deterministic counterpart almost
surely.
http://front.math.ucdavis.edu/math.DS/0612774
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4737. HOW TO CHOOSE A CHAMPION
E. Ben-Naim and N.W. Hengartner
League competition is investigated using random processes and scaling
techniques. In our model, a weak team can upset a strong team with a
fixed
probability. Teams play an equal number of head-to-head matches and
the team
with the largest number of wins is declared to be the champion. The
total
number of games needed for the best team to win the championship with
high
certainty, T, grows as the cube of the number of teams, N, i.e., T ~
N^3. This
number can be substantially reduced using preliminary rounds where
teams play a
small number of games and subsequently, only the top teams advance to
the next
round. When there are k rounds, the total number of games needed for
the best
team to emerge as champion, T_k, scales as follows, T_k ~N^(\gamma_k)
with
gamma_k=1/[1-(2/3)^(k+1)]. For example, gamma_k=9/5,27/19,81/65 for
k=1,2,3.
These results suggest an algorithm for how to infer the best team
using a
schedule that is linear in N. We conclude that league format is an
ineffective
method of determining the best team, and that sequential elimination
from the
bottom up is fair and efficient.
http://front.math.ucdavis.edu/physics/0612217
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4738. ON THE EXCURSION THEORY FOR LINEAR DIFFUSIONS
Paavo Salminen and Pierre Vallois and Marc Yor
We present a number of important identities related to the excursion
theory
of linear diffusions. In particular, excursions straddling an
independent
exponential time are studied in detail. Letting the parameter of the
exponential time tend to zero it is seen that these results connect
to the
corresponding results for excursions of stationary diffusions (in
stationary
state). We characterize also the laws of the diffusion prior and
posterior to
the last zero before the exponential time. It is proved using Krein's
representations that, e.g., the law of the length of the excursion
straddling
an exponential time is infinitely divisible. As an illustration of
the results
we discuss Ornstein-Uhlenbeck processes.
http://front.math.ucdavis.edu/math.PR/0612687
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4739. OPTION PRICING WITH LOG-STABLE L\'{E}VY PROCESSES
Przemys{\l}aw Repetowicz and Peter Richmond
We model the logarithm of the price (log-price) of a financial asset
as a
random variable obtained by projecting an operator stable random
vector with a
scaling index matrix $\underline{\underline{E}}$ onto a non-random
vector. The
scaling index $\underline{\underline{E}}$ models prices of the
individual
financial assets (stocks, mutual funds, etc.). We find the functional
form of
the characteristic function of real powers of the price returns and
we compute
the expectation value of these real powers and we speculate on the
utility of
these results for statistical inference. Finally we consider a portfolio
composed of an asset and an option on that asset. We derive the
characteristic
function of the deviation of the portfolio, \mbox{${\mathfrak D}_t^
{({\mathfrak
t})}$}, defined as a temporal change of the portfolio diminished by
the the
compound interest earned. We derive pseudo-differential equations for
the
option as a function of the log-stock-price and time and we find exact
closed-form solutions to that equation. These results were not known
before.
Finally we discuss how our solutions correspond to other approximate
results
known from literature,in particular to the well known Black & Scholes
equation.
http://front.math.ucdavis.edu/math.PR/0612691
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4740. OSCILLATIONS OF EMPIRICAL DISTRIBUTION FUNCTIONS UNDER DEPENDENCE
Wei Biao Wu
We obtain an almost sure bound for oscillation rates of empirical
distribution functions for stationary causal processes. For short-range
dependent processes, the oscillation rate is shown to be optimal in
the sense
that it is as sharp as the one obtained under independence. The
dependence
conditions are expressed in terms of physical dependence measures
which are
directly related to the data-generating mechanism of the underlying
processes
and thus are easy to work with.
http://front.math.ucdavis.edu/math.PR/0612692
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4741. KARHUNEN-LO\`{E}VE EXPANSIONS OF MEAN-CENTERED WIENER PROCESSES
Paul Deheuvels
For $\gamma>-{1/2}$, we provide the Karhunen-Lo\`{e}ve expansion of the
weighted mean-centered Wiener process, defined by \[W
_{\gamma}(t)=\frac{1}{\sqrt{1+2\gamma}}\Big\{W\big(t^{1+2\gamma}\big)-
\int_0^1W\big(u^{1+2\gamma}\big)du\Big\},\] for $t\in(0,1]$. We show
that the
orthogonal functions in these expansions have simple expressions in
term of
Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_
{\gamma}$
is identical in distribution with the $L^2[0,1]$ norm of the weighted
Brownian
bridge $t^{\gamma}B(t)$.
http://front.math.ucdavis.edu/math.PR/0612693
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4742. FRACTIONAL BROWNIAN FIELDS, DUALITY, AND MARTINGALES
Vladimir Dobri\'{c} and Francisco M. Ojeda
In this paper the whole family of fractional Brownian motions is
constructed
as a single Gaussian field indexed by time and the Hurst index
simultaneously.
The field has a simple covariance structure and it is related to two
generalizations of fractional Brownian motion known as
multifractional Brownian
motions. A mistake common to the existing literature regarding
multifractional
Brownian motions is pointed out and corrected. The Gaussian field,
due to
inherited ``duality'', reveals a new way of constructing martingales
associated
with the odd and even part of a fractional Brownian motion and
therefore of the
fractional Brownian motion. The existence of those martingales and their
stochastic representations is the first step to the study of natural
wavelet
expansions associated to those processes in the spirit of our earlier
work on a
construction of natural wavelets associated to Gaussian-Markov
processes.
http://front.math.ucdavis.edu/math.PR/0612694
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4743. A GENERALIZED OCCUPATION TIME FORMULA FOR CONTINUOUS
SEMIMARTINGALES
Raouf Ghomrasni
We show that for a wide class of functions $F$ that: $$ {\lim_{\epsilon
\downarrow 0} {\frac{1}{\epsilon}} \int_0^t \Big\{F(s, X_s) - F(s, X_s -
\epsilon)\Big\} d\big<X,X\big>_s} = - \int_0^t\int_{\R} F(s, x) d
L_s^x $$
where $X_t$ is a continuous semi-martingale, $(L_t^x, x \in \R, t
\geq 0)$ its
local time process and $(\big<X,X\big>_t, t \geq 0)$ its quadratic
variation
process.
http://front.math.ucdavis.edu/math.PR/0612699
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4744. FRACTAL PROPERTIES OF THE RANDOM STRING PROCESSES
Dongsheng Wu and Yimin Xiao
Let $\{u_t(x),t\ge 0, x\in {\mathbb{R}}\}$ be a random string taking
values
in ${\mathbb{R}}^d$, specified by the following stochastic partial
differential
equation [Funaki (1983)]: \[\frac{\partial u_t(x)}{\partial
t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},\] where $\dot{W}
(x,t)$ is
an ${\mathbb{R}}^d$-valued space-time white noise. Mueller and Tribe
(2002)
have proved necessary and sufficient conditions for the ${\mathbb{R}}
^d$-valued
process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$ to hit points and to
have double
points. In this paper, we continue their research by determining the
Hausdorff
and packing dimensions of the level sets and the sets of double times
of the
random string process $\{u_t(x):t\ge 0, x\in {\mathbb{R}}\}$. We also
consider
the Hausdorff and packing dimensions of the range and graph of the
string.
http://front.math.ucdavis.edu/math.PR/0612700
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4745. MODIFIED EMPIRICAL CLT'S UNDER ONLY PRE-GAUSSIAN CONDITIONS
Shahar Mendelson and Joel Zinn
We show that a modified Empirical process converges to the limiting
Gaussian
process whenever the limit is continuous. The modification depends on
the
properties of the limit via Talagrand's characterization of the
continuity of
Gaussian processes.
http://front.math.ucdavis.edu/math.PR/0612703
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4746. EMPIRICAL AND GAUSSIAN PROCESSES ON BESOV CLASSES
Richard Nickl
We give several conditions for pregaussianity of norm balls of Besov
spaces
defined over $\mathbb{R}^d$ by exploiting results in Haroske and Triebel
(2005). Furthermore, complementing sufficient conditions in Nickl and
P\"{o}tscher (2005), we give necessary conditions on the parameters
of the
Besov space to obtain the Donsker property of such balls. For certain
parameter
combinations Besov balls are shown to be pregaussian but not Donsker.
http://front.math.ucdavis.edu/math.PR/0612706
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4747. INVARIANCE PRINCIPLE FOR STOCHASTIC PROCESSES WITH SHORT MEMORY
Magda Peligrad and Sergey Utev
In this paper we give simple sufficient conditions for linear type
processes
with short memory that imply the invariance principle. Various examples
including projective criterion are considered as applications. In
particular,
we treat the weak invariance principle for partial sums of linear
processes
with short memory. We prove that whenever the partial sums of
innovations
satisfy the $L_p$--invariance principle, then so does the partial
sums of its
corresponding linear process.
http://front.math.ucdavis.edu/math.PR/0612707
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4748. HOMOGENENOUS MULTITYPE FRAGMENTATIONS
Jean Bertoin (PMA and DMA)
A homogeneous mass-fragmentation, as it has been defined in \cite{RFC},
describes the evolution of the collection of masses of fragments of
an object
which breaks down into pieces as time passes. Here, we show that this
model can
be enriched by considering also the types of the fragments, where a
type may
represent, for instance, a geometrical shape, and can take finitely many
values. In this setting, the dynamics of a randomly tagged fragment
play a
crucial role in the analysis of the fragmentation. They are
determined by a
Markov additive process whose distribution depends explicitly on the
characteristics of the fragmentation. As applications, we make
explicit the
connexion with multitype branching random walks, and obtain multitype
analogs
of the pathwise central limit theorem and large deviation estimates
for the
empirical distribution of fragments.
http://front.math.ucdavis.edu/math.PR/0612710
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4749. PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE
Adrian P.C. Lim
A typical path integral on a manifold, $M$ is an informal expression
of the
form \frac{1}{Z}\int_{\sigma \in H(M)} f(\sigma)
e^{-E(\sigma)}\mathcal{D}\sigma, \nonumber where $H(M)$ is a Hilbert
manifold
of paths with energy $E(\sigma) < \infty$, $f$ is a real valued
function on
$H(M)$, $\mathcal{D}\sigma$ is a \textquotedblleft Lebesgue measure
\textquotedblright and $Z$ is a normalization constant. For a compact
Riemannian manifold $M$, we wish to interpret $\mathcal{D}\sigma$ as a
Riemannian \textquotedblleft volume form \textquotedblright over $H(M)$,
equipped with its natural $G^{1}$ metric. Given an equally spaced
partition,
${\mathcal{P}}$ of $[0,1],$ let $H_{{\mathcal{P}}%}(M)$ be the finite
dimensional Riemannian submanifold of $H(M) $ consisting of piecewise
geodesic
paths adapted to $\mathcal{P.}$ Under certain curvature restrictions
on $M,$ it
is shown that \[
\frac{1}{Z_{{\mathcal{P}}}}e^{-{1/2}E(\sigma)}dVol_{H_{{\mathcal{P}}}%
}(\sigma)\to\rho(\sigma)d\nu(\sigma)\text{as}\mathrm{mesh}%
({\mathcal{P}})\to0, \] where $Z_{{\mathcal{P}}}$ is a \textquotedblleft
normalization\textquotedblright constant, $E:H(M) \to\lbrack0,\infty)
$ is the
energy functional, $Vol_{H_{{\mathcal{P}}%}}$ is the Riemannian
volume measure
on $H_{\mathcal{P}}(M) ,$ $\nu$ is Wiener measure on continuous paths
in $M,$
and $\rho$ is a certain density determined by the curvature tensor of
$M.$
http://front.math.ucdavis.edu/math.PR/0612711
---------------------------------------------------------------
4750. RISK BOUNDS FOR THE NON-PARAMETRIC ESTIMATION OF L\'{E}VY
PROCESSES
Jos\'{e} E. Figueroa-L\'{o}pez and Christian Houdr\'{e}
Estimation methods for the L\'{e}vy density of a L\'{e}vy process are
developed under mild qualitative assumptions. A classical model
selection
approach made up of two steps is studied. The first step consists in the
selection of a good estimator, from an approximating (finite-
dimensional)
linear model ${\mathcal{S}}$ for the true L\'{e}vy density. The
second is a
data-driven selection of a linear model ${\mathcal{S}}$, among a given
collection $\{{\mathcal{S}}_m\}_{m\in {\mathcal{M}}}$, that
approximately
realizes the best trade-off between the error of estimation within
${\mathcal{S}}$ and the error incurred when approximating the true L
\'{e}vy
density by the linear model ${\mathcal{S}}$. Using recent concentration
inequalities for functionals of Poisson integrals, a bound for the
risk of
estimation is obtained. As a byproduct, oracle inequalities and long-run
asymptotics for spline estimators are derived. Even though the resulting
underlying statistics are based on continuous time observations of
the process,
approximations based on high-frequency discrete-data can be easily
devised.
http://front.math.ucdavis.edu/math.ST/0612697
---------------------------------------------------------------
4751. REVISITING TWO STRONG APPROXIMATION RESULTS OF DUDLEY AND PHILIPP
Philippe Berthet and David M. Mason
We demonstrate the strength of a coupling derived from a Gaussian
approximation of Zaitsev (1987a) by revisiting two strong
approximation results
for the empirical process of Dudley and Philipp (1983), and using the
coupling
to derive extended and refined versions of them.
http://front.math.ucdavis.edu/math.ST/0612701
---------------------------------------------------------------
4752. ON THE BAHADUR SLOPE OF THE LILLIEFORS AND THE CRAM\'{E}R--VON
MISES TESTS OF NORMALITY
Miguel A. Arcones
We find the Bahadur slope of the Lilliefors and Cram\'{e}r--von Mises
tests
of normality.
http://front.math.ucdavis.edu/math.ST/0612708
---------------------------------------------------------------
4753. IVY ON THE CEILING: FIRST-ORDER POLYMER DEPINNING TRANSITIONS
WITH QUENCHED DISORDER
Kenneth S. Alexander
We consider a polymer, with monomer locations modeled by the
trajectory of an
underlying Markov chain, in the presence of a potential thatinteracts
with the
polymer when it visits a particular site 0. Disorder is introduced by
having
the interaction vary from one monomer to another, as a constant $u$
plus i.i.d.
mean-0 randomness. There is a critical value of $u$ above which the
polymer is
pinned, placing a positive fraction (called the contact fraction) of its
monomers at 0 with high probability. When the excursions of the
underlying
chain have a finite mean but no finite exponential moment, it is
known that the
depinning transition (more precisely, the contact fraction) in the
corresponding annealed system is discontinuous. One generally expects
the
presence of disorder to smooth transitions, and it was proved by
Giacomin and
Toninelli that when the excursion length distribution has power-law
tails, the
quenched system has a continuous transition even if the annealed
system does
not. We show here that when the underlying chain is transient but the
finite
part of the excursion length distribution has exponential tails, then
the
depinning transition is discontinuous even in the quenched system,
and the
quenched and annealed critical points are strictly different. By
contrast, in
the recurrent case, the depinning behavior depends on the subexponential
prefactors on the exponential decay of the excursion length
distribution, and
when these prefactors decay with an appropriate power law, the quenched
transition is continuous even though the annealed one is not.
http://front.math.ucdavis.edu/math.PR/0612625
---------------------------------------------------------------
4754. MERGING PERCOLATION ON $Z^D$ AND CLASSICAL RANDOM GRAPHS:
PHASE TRANSITION
Tatyana S. Turova and Thomas Vallier
We study a random graph model which is a superposition of the bond
percolation model on $Z^d$ with probability $p$ of an edge, and a
classical
random graph $G(n, c/n)$. We show that this model, being a {\it
homogeneous}
random graph, has a natural relation to the so-called "rank 1 case"
of {\it
inhomogeneous} random graphs. This allows us to use the newly
developed theory
of inhomogeneous random graphs to describe the phase diagram on the
set of
parameters $c\geq 0$ and $0 \leq p<p_c$, where $p_c=p_c(d)$ is the
critical
probability for the bond percolation on $Z^d$. The phase transition
is similar
to the classical random graph, it is of the second order. We also
find the
scaled size of the largest connected component above the phase
transition.
http://front.math.ucdavis.edu/math.PR/0612644
---------------------------------------------------------------
4755. A CALL-PUT DUALITY FOR PERPETUAL AMERICAN OPTIONS
Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
It is well known that in models with time-homogeneous local volatility
functions and constant interest and dividend rates, the European Put
prices are
transformed into European Call prices by the simultaneous exchanges
of the
interest and dividend rates and of the strike and spot price of the
underlying.
This paper investigates such a Call Put duality for perpetual
American options.
It turns out that the perpetual American Put price is equal to the
perpetual
American Call price in a model where, in addition to the previous
exchanges
between the spot price and the strike and between the interest and
dividend
rates, the local volatility function is modified. We prove that
equality of the
dual volatility functions only holds in the standard Black-Scholes
model with
constant volatility. Thanks to these duality results, we design a
theoretical
calibration procedure of the local volatility function from the
perpetual Call
and Put prices for a fixed spot price $x_0$. The knowledge of the Put
(resp.
Call) prices for all strikes enables to recover the local volatility
function
on the interval $(0,x_0)$ (resp. $(x_0,+\infty)$).
http://front.math.ucdavis.edu/math.PR/0612648
---------------------------------------------------------------
4756. GENERAL DUALITY FOR PERPETUAL AMERICAN OPTIONS
Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
In this paper, we investigate the generalization of the Call-Put duality
equality obtained in [1] for perpetual American options when the Call-
Put
payoff $(y-x)^+$ is replaced by $\phi(x,y)$. It turns out that the
duality
still holds under monotonicity and concavity assumptions on $\phi$. The
specific analytical form of the Call-Put payoff only makes
calculations easier
but is not crucial unlike in the derivation of the Call-Put duality
equality
for European options. Last, we give some examples for which the optimal
strategy is known explicitly.
http://front.math.ucdavis.edu/math.PR/0612649
---------------------------------------------------------------
4757. LEVEL CROSSINGS AND OTHER LEVEL FUNCTIONALS OF STATIONARY
GAUSSIAN PROCESSES
Marie F. Kratz
This paper presents a synthesis on the mathematical work done on level
crossings of stationary Gaussian processes, with some extensions. The
main
results [(factorial) moments, representation into the Wiener Chaos,
asymptotic
results, rate of convergence, local time and number of crossings] are
described, as well as the different approaches [normal comparison
method, Rice
method, Stein-Chen method, a general $m$-dependent method] used to
obtain them;
these methods are also very useful in the general context of Gaussian
fields.
Finally some extensions [time occupation functionals, number of
maxima in an
interval, process indexed by a bidimensional set] are proposed,
illustrating
the generality of the methods. A large inventory of papers and books
on the
subject ends the survey.
http://front.math.ucdavis.edu/math.PR/0612577
---------------------------------------------------------------
4758. A RANDOM MULTIPLE ACCESS PROTOCOL WITH SPATIAL INTERACTIONS
Charles Bordenave and Serguei Foss and Vsevolod Shneer
We analyse an aloha type access protocol where users have local
interactions.
We establish that the fluid model of the system workload satisfies a
differential equation. We exhibit a sufficient condition on the
stability of
this differential equation and deduce a sufficient condition for the
stability
of the protocol. We discuss the necessary condition.
http://front.math.ucdavis.edu/math.PR/0612583
---------------------------------------------------------------
4759. VOLUME GROWTH AND HEAT KERNEL ESTIMATES FOR THE CONTINUUM
RANDOM TREE
David Croydon
In this article, we prove global and local (point-wise) volume and heat
kernel bounds for the continuum random tree. We demonstrate that
there are
almost-surely logarithmic global fluctuations and log-logarithmic local
fluctuations in the volume of balls of radius $r$ about the leading
order
polynomial term as $r\to0$. We also show that the on-diagonal part of
the heat
kernel exhibits corresponding global and local fluctuations as $t\to0$
almost-surely. Finally, we prove that this quenched (almost-sure)
behaviour
contrasts with the local annealed (averaged over all realisations of
the tree)
volume and heat kernel behaviour, which is smooth.
http://front.math.ucdavis.edu/math.PR/0612585
---------------------------------------------------------------
4760. STOCHASTIC INTEGRALS AND ASYMPTOTIC ANALYSIS OF CANONICAL VON
MISES STATISTICS BASED ON DEPENDENT OBSERVATIONS
Igor S. Borisov and Alexander A. Bystrov
In the first part of the paper we study stochastic integrals of a
nonrandom
function with respect to a nonorthogonal Hilbert noise defined on a
semiring of
subsets of an arbitrary nonempty set. In the second part we apply this
construction to study limit behavior of canonical (i.e., degenerate)
Von Mises
statistics based on weakly dependent stationary observations.
http://front.math.ucdavis.edu/math.PR/0612594
---------------------------------------------------------------
4761. THE LENGTH OF AN SLE - MONTE CARLO STUDIES
Tom Kennedy
The scaling limits of a variety of critical two-dimensional lattice
models
are equal to the Schramm-Loewner evolution (SLE) for a suitable value
of the
parameter kappa. These lattice models have a natural parametrization
of their
random curves given by the length of the curve. This parametrization
(with
suitable scaling) should provide a natural parametrization for the
curves in
the scaling limit. We conjecture that this parametrization is also
given by a
type of fractal variation along the curve, and present Monte Carlo
simulations
to support this conjecture. Then we show by simulations that if this
fractal
variation is used to parametrize the SLE, then the parametrized
curves have the
same distribution as the curves in the scaling limit of the lattice
models with
their natural parametrization.
http://front.math.ucdavis.edu/math.PR/0612609
---------------------------------------------------------------
4762. LINEAR RAMSEY NUMBERS FOR BOUNDED-DEGREE HYPERGRAPHS
Yoshiyasu Ishigami
We show that the Ramsey number is linear for every uniform hypergraph
with
bounded-degree. This is a hypergraph extension of the famous theorem for
ordinary graphs which Chv\'atal et al. showed in 1983. Our proof is
simple,
contains the multicolor case, and provides a strong embedding lemma.
http://front.math.ucdavis.edu/math.CO/0612601
---------------------------------------------------------------
4763. GAUSSIAN PROCESSES, KINEMATIC FORMULAE AND POINCAR\'E'S LIMIT
Jonathan E. Taylor and Robert J. Adler
We consider vector valued, unit variance Gaussian processes $y$
defined over
piecewise $C^2$ stratified manifolds $M$ and consider the geometry of
their
(random) excursion sets $M\cap y^{-1}D$ for $D$ a stratified subset of
Euclidean space. In particular, we develop an explicit formula for the
expectation of all the Lipshitz-Killing curvatures of these sets.
This formula
has an interpretation as a version of the classic kinematic
fundamental formula
of Integral Geometry, in which integration over the isometry group
with respect
to Haar measure is replaced by integration over a function space with
respect
to an appropriate Gaussian measure.
Particularly novel is the method of proof, based on approximating the
Gaussian processes by processes on spheres, the orthonormal
expansions of which
have (random) coefficients on the $n$-sphere. The $n\to\infty$ limit
is handled
via recent extensions of the classic Poincar\'e limit theorem.
http://front.math.ucdavis.edu/math.DG/0612580
---------------------------------------------------------------
4764. THE ASYMPTOTIC BEHAVIOS OF FREE CONVOLUTION
Hari Bercovici and Jiun-Chau Wang
We give a streamlined proof of the limit theorems for the free additive
convolution of infinitesimal triangular arrays of probability
measures on the
real line. The result was first proved by Chistyakov and G\"otze
using analytic
subordination.
http://front.math.ucdavis.edu/math.OA/0612599
---------------------------------------------------------------
4765. LOCALIZATION OF FAVORITE POINTS FOR DIFFUSION IN RANDOM
ENVIRONMENT
Dimitrios Cheliotis
For a diffusion X_t in a one-dimensional Wiener medium W, it is known
that
there is a certain process b_x(W) that depends only on the
environment W, so
that X_t-b_{logt}(W) converges in distribution as t goes to infinity.
We prove
that, modulo a relatively small time change, the process {b_x(W):x>0}is
followed closely by the process {F_X(e^x): x>0}, with F_X(t) denoting
the point
with the most local time for the diffusion at time t.
http://front.math.ucdavis.edu/math.PR/0612533
---------------------------------------------------------------
4766. RANK DISTRIBUTIONS IN SEMIOTICS
V. P. Maslov and T. V. Maslova
The notions of real and user cardinality of a sign are introduced. Rank
distributions can be extended to arbitrary sign objects, i.e., semiotic
systems. The dynamics of the distribution of consumer durables, such as
automobiles, is studied.
http://front.math.ucdavis.edu/math.PR/0612540
---------------------------------------------------------------
4767. ON EXPONENTIAL ERGODICITY OF MULTICLASS QUEUEING NETWORKS
David Gamarnik and Sean Meyn
One of the key performance measures in queueing systems is the
exponential
decay rate of the steady-state tail probabilities of the queue
lengths. It is
known that if a corresponding fluid model is stable and the stochastic
primitives have finite moments, then the queue lengths also have finite
moments, so that the tail probability \pr(\cdot >s) decays faster
than s^{-n}
for any n. It is natural to conjecture that the decay rate is in fact
exponential.
In this paper an example is constructed to demonstrate that this
conjecture
is false. For a specific stationary policy applied to a network with
exponentially distributed interarrival and service times it is shown
that the
corresponding fluid limit model is stable, but the tail probability
for the
buffer length decays slower than s^{-\log s}.
http://front.math.ucdavis.edu/math.PR/0612544
---------------------------------------------------------------
4768. A CONTACT PROCESS WITH MUTATIONS ON A TREE
Thomas M. Liggett and Rinaldo B. Schinazi and and Jason Schweinsberg
Consider the following stochastic model for immune response. Each
pathogen
gives birth to a new pathogen at rate $\lambda$. When a new pathogen
is born,
it has the same type as its parent with probability $1 - r$. With
probability
$r$, a mutation occurs, and the new pathogen has a different type
from all
previously observed pathogens. When a new type appears in the
population, it
survives for an exponential amount of time with mean 1, independently
of all
the other types. All pathogens of that type are killed
simultaneously. Schinazi
and Schweinsberg (2006) have shown that this model on $\Z^d$ behaves
rather
differently from its non-spatial version. In this paper, we show that
this
model on a homogeneous tree captures features from both the non-
spatial version
and the $\Z^d$ version. We also obtain comparison results between
this model
and the basic contact process on general graphs.
http://front.math.ucdavis.edu/math.PR/0612564
---------------------------------------------------------------
4769. A NEW APPROACH FOR CAPACITY ANALYSIS OF LARGE DIMENSIONAL MULTI-
ANTENNA CHANNELS
Walid Hachem (LTCI) and Oleksiy Khorunzhiy and Philippe Loubaton
(IGM-LabInfo), Jamal Najim (LTCI), Leonid Pastur
This paper adresses the behaviour of the mutual information of
correlated
MIMO Rayleigh channels when the numbers of transmit and receive antennas
converge to infinity at the same rate. Using a new and simple
approach based on
Poincar\'{e}-Nash inequality and on an integration by parts formula,
it is
rigorously established that the mutual information converges to a
Gaussian
random variable whose mean and variance are evaluated. These results
confirm
previous evaluations based on the powerful but non rigorous replica
method. It
is believed that the tools that are used in this paper are simple,
robust, and
of interest for the communications engineering community.
http://front.math.ucdavis.edu/cs.IT/0612076
---------------------------------------------------------------
4770. A NEW METHOD FOR QUEUING PERFORMANCE ESTIMATES USING MARKOV CHAINS
Richard G. Clegg
This paper gives an exact closed form solution for the expected queue
length
at equilibrium of a G/D/1 discrete time queuing system in which the
arrival
process is a specific Markov-modulated process. A system of equations
is given
which can calculate the probability that the queue has a given
length. The
results are tested in simulation.
http://front.math.ucdavis.edu/math.PR/0612476
---------------------------------------------------------------
4771. THE AREA OF EXPONENTIAL RANDOM WALK AND PARTIAL SUMS OF UNIFORM
ORDER STATISTICS
Vladislav Vysotsky
Let S_i be a random walk with standard exponential increments. We call
\sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1}
\frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of
so-called one-dimensional sticky particles model. We find the
distribution of V
and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We
also show
that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}
^k U_{i, n}
converge in distribution to V. Here U_{i, n} are the order statistics
of n
i.i.d. random variables uniformly distributed on [0,1].
http://front.math.ucdavis.edu/math.PR/0612490
---------------------------------------------------------------
4772. MULTI-STEP RICHARDSON-ROMBERG EXTRAPOLATION: REMARKS ON
VARIANCE CONTROL AND COMPLEXITY
Gilles Pag\`{e}s (PMA)
We propose a multi-step Richardson-Romberg extrapolation method for the
computation of expectations $E f(X_{_T})$ of a diffusion $(X_t)_{t\in
[0,T]}$
when the weak time discretization error induced by the Euler scheme
admits an
expansion at an order $R\ge 2$. The complexity of the estimator grows
as $R^2$
(instead of $2^R$) and its variance is asymptotically controlled by
considering
some consistent Brownian increments in the underlying Euler schemes.
Some Monte
carlo simulations carried with path-dependent options (lookback,
barriers)
which support the conjecture that their weak time discretization
error also
admits an expansion (in a different scale). Then an appropriate
Richardson-Romberg extrapolation seems to outperform the Euler scheme
with
Brownian bridge.
http://front.math.ucdavis.edu/math.PR/0612523
---------------------------------------------------------------
4773. CAPITAL ALLOCATION FOR CREDIT PORTFOLIOS WITH KERNEL ESTIMATORS
Dirk Tasche
Determining contributions by sub-portfolios or single exposures to
portfolio-wide economic capital for credit risk is an important risk
measurement task. Often economic capital is measured as Value-at-Risk
(VaR) of
the portfolio loss distribution. For many of the credit portfolio
risk models
used in practice, then the VaR contributions have to be estimated
from Monte
Carlo samples. In the context of a partly continuous loss
distribution (i.e.
continuous except for a positive point mass on zero), we investigate
how to
combine kernel estimation methods with importance sampling to achieve
more
efficient (i.e. less volatile) estimation of VaR contributions.
http://front.math.ucdavis.edu/math.ST/0612470
---------------------------------------------------------------
4774. ON THE HYPERPLANE CONJECTURE FOR RANDOM CONVEX SETS
Bo'az Klartag and Gady Kozma
Let N > n, and denote by K the convex hull of N independent standard
gaussian
random vectors in an n-dimensional Euclidean space. We prove that
with high
probability, the isotropic constant of K is bounded by a universal
constant.
Thus we verify the hyperplane conjecture for the class of gaussian
random
polytopes.
http://front.math.ucdavis.edu/math.MG/0612517
---------------------------------------------------------------
4775. A NEW REM CONJECTURE
Gerard Ben Arous and Veronique Gayrard and Alexey Kuptsov
We introduce here a new universality conjecture for levels of random
Hamiltonians, in the same spirit as the local REM conjecture made by
S. Mertens
and H. Bauke. We establish our conjecture for a wide class of
Gaussian and
non-Gaussian Hamiltonians, which include the $p$-spin models, the
Sherrington-Kirkpatrick model and the number partitioning problem. We
prove
that our universality result is optimal for the last two models by
showing when
this universality breaks down.
http://front.math.ucdavis.edu/math.PR/0612373
---------------------------------------------------------------
4776. TIGHTNESS FOR A FAMILY OF RECURSIVE EQUATIONS
Maury Bramson and Ofer Zeitouni
In this paper, we study the tightness of solutions for a family of
recursive
equations. These equations arise naturally in the study of random
walks on
tree-like structures. Examples include the maximal displacement of
branching
random walk in one dimension, and the cover time of symmetric simple
random
walk on regular binary trees. Recursion equations associated with the
distribution functions of these quantities have been used to
establish weak
laws of large numbers. Here, we use these recursion equations to
establish the
tightness of the corresponding sequences of distribution functions after
appropriate centering. We phrase our results in a fairly general
context, which
we hope will facilitate their application in other settings.
http://front.math.ucdavis.edu/math.PR/0612382
---------------------------------------------------------------
4777. JOINT PROBABILITY FOR THE PEARCEY PROCESS
Mark Adler & Pierre van Moerbeke
This paper is a step in the direction of understanding the behavior of
non-intersecting Brownian motions on the real line, when the number of
particles becomes large.
Consider 2k non-intersecting Brownian motions, all starting at the
origin,
such that the k left paths end up at -a and the k right paths end up
at +a at
time t=1. The Karlin-McGregor formula enables one to express the
transition
probability in terms of a matrix model, consisting of Gaussian
Hermitian random
matrices in a chain with external source. It is shown that the log of
the
probability for this model satisfies a fourth order PDE with a quartic
non-linearity, obtained by means of the 3-component KP hierarchy and
Virasoro
constraints.
When the number of particles grows very large, the particles will be
concentrated on two intervals near t=0 and on one interval near t=1. The
Pearcey process is the infinite-dimensional diffusion, near the critical
transition from two to one interval. An appropriate scaling limit of
the PDE
for the finite model leads to a non-linear PDE for the multi-time
transition
probabilities of the Pearcey process.
We conjecture that each of the Markov clouds (like the Pearcey
process)
arising near phase transitions is related to some integrable system.
Moreover,
there is an intimate connection between the integrable system and the
associated Riemann-Hilbert problem.
http://front.math.ucdavis.edu/math.PR/0612393
---------------------------------------------------------------
4778. ON A DISTRIBUTION IN FREQUENCY PROBABILITY THEORY CORRESPONDING
TO THE BOSE-EINSTEIN DISTRIBUTION
V. P. Maslov
The notion of density of a finite set is discussed. We proof a general
theorem of set theory which refines Bose-Einstein distribution.
http://front.math.ucdavis.edu/math.PR/0612394
---------------------------------------------------------------
4779. INFLUENCES AND DECISION TREES
Hamed Hatami
A celebrated theorem of Friedgut says that every function $f:\{0,1\}
^n \to
\{0,1\}$ can be approximated by a function $g:\{0,1\}^n \to \{0,1\}$
with
$\|f-g\|_2^2 \le \epsilon$ which depends only on $e^{O(I_f/\epsilon)}$
variables where $I_f$ is the sum of the influences of the variables
of $f$.
Dinur and Friedgut later showed that this statement also holds if we
replace
the discrete domain $\{0,1\}^n$ with the continuous domain $[0,1]^n$,
under the
extra assumption that $f$ is monotone. They conjectured that the
condition of
monotonicity is unnecessary and can be removed.
We show that certain constant-depth decision trees provide counter-
examples
to Dinur-Friedgut conjecture. This suggests a reformulation of the
conjecture
in which the function $g:[0,1]^n \to \{0,1\}$ instead of depending on
a small
number of variables has a decision tree of small depth. In fact we
prove this
reformulation by showing that the depth of the decision tree of $g$
can be
bounded by $e^{O(I_f/\epsilon^2)}$.
http://front.math.ucdavis.edu/math.PR/0612405
---------------------------------------------------------------
4780. DYNAMICAL PROPERTIES AND CHARACTERIZATION OF GRADIENT DRIFT
DIFFUSIONS
S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA)
We study the dynamical properties of the Brownian diffusions having $
\sigma
{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift
vector. We
characterize this class through the equality $D^2_+=D^2_-$, where $D_
{+}$
(resp. $D_-$) denotes the forward (resp. backward) stochastic
derivative of
Nelson's type. Our proof is based on a remarkable identity for $D_+^2-
D_-^2$
and on the use of the martingale problem. We also give a new
formulation of a
famous theorem of Kolmogorov concerning reversible diffusions. We
finally
relate our characterization to some questions about the complex
stochastic
embedding of the Newton equation which initially motivated of this work.
http://front.math.ucdavis.edu/math.PR/0612413
---------------------------------------------------------------
4781. AN L2 THEORY FOR DIFFERENTIAL FORMS ON PATH SPACES I
K.D. Elworthy and Xue-Mei Li
An L2 theory of differential forms is proposed for the Banach
manifold of
continuous paths on Riemannian manifolds M furnished with its
Brownian motion
measure. Differentiation must be restricted to certain Hilbert space
directions, the H-tangent vectors. To obtain a closed exterior
differential
operator the relevant spaces of differential forms, the H-forms, are
perturbed
by the curvature of M. A Hodge decomposition is given for L2 H-one-
forms, and
the structure of H-two -forms is described. The dual operator d* is
analysed in
terms of a natural connection on the H-tangent spaces. Malliavin
calculus is a
basic tool.
http://front.math.ucdavis.edu/math.PR/0612416
---------------------------------------------------------------
4782. HEAT KERNEL AND GREEN FUNCTION ESTIMATES ON AFFINE BUILDINGS OF
TYPE $\TILDE{A}_R$
Jean-Philippe Anker (MAPMO) and Bruno Schapira (MAPMO and PMA) and
Bartosz Trojan (MAPMO)
We obtain a global estimate of the transition density $p^n(0,x)$
associated
to a nearest neighbor random walk, called here "simple", on affine
buildings of
type $\widetilde{A}_r$. Then we deduce a global estimate of the Green
function.
This is the analogue of a result on Riemannian symmetric spaces of the
noncompact type.
http://front.math.ucdavis.edu/math.CA/0612385
---------------------------------------------------------------
4783. SHARP THRESHOLDS FOR CONSTRAINT SATISFACTION PROBLEM AND GRAPH
HOMOMORPHISMS
Hamed Hatami and Michael Molloy
We determine under which conditions certain natural models of random
constraint satisfaction problems have sharp thresholds of
satisfiability. These
models include graph and hypergraph homomorphism, the $(d,k,t)$-
model, and
binary constraint satisfaction problems with domain size 3.
http://front.math.ucdavis.edu/math.CO/0612391
---------------------------------------------------------------
4784. ON THE MINIMIZATION OF OPERATIONAL RISKS
V. P. Maslov
We give a risk-minimizing formula for government investments taking into
account the zero intelligence law for financial markets.
http://front.math.ucdavis.edu/math.GM/0612395
---------------------------------------------------------------
4785. CROSSING PROBABILITIES FOR DIFFUSION PROCESSES WITH PIECEWISE
CONTINUOUS BOUNDARIES
Liqun Wang and Klaus P\"otzelberger
We propose an approach to compute the boundary crossing probabilities
for a
class of diffusion processes which can be expressed as piecewise
monotone (not
necessarily one-to-one) functionals of a standard Brownian motion.
This class
includes many interesting processes in real applications, e.g.,
Ornstein-Uhlenbeck, growth processes and geometric Brownian motion
with time
dependent drift. This method applies to both one-sided and two-sided
general
nonlinear boundaries, which may be discontinuous. Using this approach
explicit
formulas for boundary crossing probabilities for certain nonlinear
boundaries
are obtained, which are useful in evaluation and comparison of various
omputational algorithms. Moreover, numerical computation can be
easily done by
Monte Carlo integration and the approximation errors for general
boundaries are
automatically calculated. Some numerical examples are presented.
http://front.math.ucdavis.edu/math.PR/0612337
---------------------------------------------------------------
4786. WHAT IS THE NATURAL SCALE FOR A L\'EVY PROCESS IN MODELLING
TERM STRUCTURE OF INTEREST RATES?
Jir\^o Akahori and Takahiro Tsuchiya
This paper gives examples of explicit arbitrage-free term structure
models
with L\'evy jumps via state price density approach. By generalizing
quadratic
Gaussian models, it is found that the probability density function of
a L\'evy
process is a "natural" scale for the process to be the state variable
of a
market.
http://front.math.ucdavis.edu/math.PR/0612341
---------------------------------------------------------------
4787. ON THE LONGEST INCREASING SUBSEQUENCE FOR FINITE AND COUNTABLE
ALPHABETS
Christian houdr\'e and Trevis J. Litherland
Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables
with
values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length
of the
longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express
the limiting
distribution of $LI_n$ as functionals of $m$ and $(m-1)$-dimensional
Brownian
motions. These expressions are then related to similar functionals
appearing in
queueing theory, allowing us to further establish asymptotic
behaviors as $m$
grows. The finite alphabet results are then used to treat the countable
(infinite) alphabet.
http://front.math.ucdavis.edu/math.PR/0612364
---------------------------------------------------------------
4788. ANTICIPATING REFLECTED STOCHASTIC DIFFERENTIAL EQUATIONS
Zongxia Liang and Tusheng Zhang
In this paper, we establish the existence of the solutions $ (X, L)$ of
reflected stochastic differential equations with possible
anticipating initial
random variables. The key is to obtain some substitution formula for
Stratonovich integrals via a uniform convergence of the corresponding
Riemann
sums.
http://front.math.ucdavis.edu/math.PR/0612294
---------------------------------------------------------------
4789. ON RECURRENCE OF REFLECTED RANDOM WALK ON THE HALF-LINE. WITH
AN APPENDIX ON RESULTS OF MARTIN BENDA
Marc Peign\'e and Wolfgang Woess
Let $(Y_n)$ be a sequence of i.i.d. real valued random variables.
Reflected
random walk $(X_n)$ is defined recursively by $X_0=x \ge 0$, $X_{n+1}
= |X_n -
Y_{n+1}|$. In this note, we study recurrence of this process,
extending a
previous criterion. This is obtained by determining an invariant
measure of the
embedded process of reflections.
http://front.math.ucdavis.edu/math.PR/0612306
---------------------------------------------------------------
4790. ON A MODEL FOR THE STORAGE OF FILES ON A HARDWARE II :
EVOLUTION OF A TYPICAL DATA BLOCK
Vincent Bansaye (PMA)
We consider a generalized version in continuous time of the parking
problem
of Knuth. Files arrive following a Poisson point process and are
stored on a
hardware identified with the real line, at the right of their arrival
point. We
study here the evolution of the extremities of the data block
straddling 0,
which is empty at time 0 and is equal to $\RRR$ at a deterministic time.
http://front.math.ucdavis.edu/math.PR/0612312
---------------------------------------------------------------
4791. FREE-KNOT SPLINE APPROXIMATION OF STOCHASTIC PROCESSES
J. Creutzig and T. Mueller-Gronbach and K. Ritter
We study optimal approximation of stochastic processes by polynomial
splines
with free knots. The number of free knots is either a priori fixed or
may
depend on the particular trajectory. For the $s$-fold integrated
Wiener process
as well as for scalar diffusion processes we determine the asymptotic
behavior
of the average $L_p$-distance to the splines spaces, as the
(expected) number
$k$ of free knots tends to infinity.
http://front.math.ucdavis.edu/math.PR/0612313
---------------------------------------------------------------
4792. SCALING LIMITS OF BIPARTITE PLANAR MAPS ARE HOMEOMORPHIC TO THE
2-SPHERE
J.F. Le Gall and F. Paulin
We prove that scaling limits of random planar maps which are uniformly
distributed over the set of all rooted 2k-angulations are a.s.
homeomorphic to
the two-dimensional sphere. Our methods rely on the study of certain
random
geodesic laminations of the disk.
http://front.math.ucdavis.edu/math.PR/0612315
---------------------------------------------------------------
4793. AUTOMORPHISMS OF THE TYPE II_1 ARVESON SYSTEM OF WARREN'S NOISE
Boris Tsirelson
Motions of the plane (shifts and rotations) correspond to
automorphisms of
the type I Arveson system of white noise. I prove that automorphisms
corresponding to rotations cannot be extended to the type II Arveson
system of
Warren's noise.
http://front.math.ucdavis.edu/math.OA/0612303
---------------------------------------------------------------
4794. A LARGE CLOSED QUEUEING NETWORK IN MARKOV ENVIRONMENT AND ITS
APPLICATION
Vyacheslav M. Abramov
A paper studies a closed queueing network containing a server station
and $k$
client stations. The server station is an infinite server queueing
system, and
client stations are single server queueing systems with autonomous
service,
i.e. every client station serves customers (units) only at random
instants
generated by strictly stationary and ergodic sequence of random
variables. The
total number of units in the network is $N$. The expected times between
departures in client stations are $(N\mu_j)^{-1}$. After service
completion in
the server station a unit is transmitted to the $j$th client station
with
probability $p_{j}$ $(j=1,2,...,k)$, and being processed in the $j$th
client
station the unit returns to server station. The network is assumed to
be in
Markov environment. The Markov environment is defined by initial
state, and
phase space of dimension $d$. Then the routing matrix $p_{j}$ as well as
transmission rates (which are expressed via parameters of the
network) depend
on the Markov state of the environment. The paper studies the queue-
length
processes in client stations of this network, and is aimed to
analysis of
performance measures associated with this network. The questions
risen in this
paper have immediate relation to quality control of complex
telecommunication
networks.
http://front.math.ucdavis.edu/math.PR/0612224
---------------------------------------------------------------
4795. A FUNCTIONAL LIMIT THEOREM FOR THE POSITION OF A PARTICLE IN A
LORENTZ TYPE MODEL
Vladislav Vysotsky
Consider a particle moving through a random medium, which consists of
spherical obstacles, randomly distributed in R^d. The particle is
accelerated
by a constant external field; when colliding with an obstacle, the
particle
inelastically reflects. We study the asymptotics of X(t), which
denotes the
position of the particle at time t, as t tends to infinity. The
result is a
functional limit theorem for X(t).
http://front.math.ucdavis.edu/math.PR/0612253
---------------------------------------------------------------
4796. THE POISSON BOUNDARY OF TRIANGULAR MATRICES IN A NUMBER FIELD
Bruno Schapira (MAPMO and PMA)
The aim of this note is to describe the Poisson boundary of the group of
invertible triangular matrices with coefficients in a number field. It
generalizes to any dimension and to any number field a result of
Brofferio
\cite{Bro} concerning the Poisson boundary of random rational
affinities.
http://front.math.ucdavis.edu/math.PR/0612272
---------------------------------------------------------------
4797. ERROR STRUCTURES AND PARAMETER ESTIMATION
Nicolas Bouleau (CERMICS) and Christophe Chorro (CERMICS and CERMSEM)
This article proposes a link between statistics and the theory of
Dirichlet
forms used to compute errors. The error calculus based on Dirichlet
forms is an
extension of classical Gauss' approach to error propagation. The aim
of this
paper is to derive error structures from measurements. The links with
Fisher's
information lay the foundations of a strong connection with
experiment. We show
that this connection behaves well towards changes of variables and is
related
to the theory of asymptotic statistics.
http://front.math.ucdavis.edu/math.ST/0612258
---------------------------------------------------------------
4798. ON MIXING AND ERGODICITY IN LOCALLY COMPACT MOTION GROUPS
M. Anoussis and D. Gatzouras
Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian
and $K$
compact. We characterize spread-out probability measures on $G$ that
are mixing
by convolutions by means of their Fourier transforms. A key tool is a
spectral
radius formula for the Fourier transform of a regular Borel measure
on $G$ that
we develop, and which is analogous to the well-known Beurling--
Gelfand spectral
radius formula. For spread-out probability measures on $G$, we also
characterize ergodicity by means of the Fourier transform of the
measure.
Finally, we show that spread-out probability measures on such groups
are mixing
if and only if they are weakly mixing.
http://front.math.ucdavis.edu/math.FA/0612262
---------------------------------------------------------------
4799. LIMIT THEOREMS FOR FREE MULTIPLICATIVE CONVOLUTIONS
Hari Bercovici and Jiun-Chau Wang
We determine the distributional behavior for products of free random
variables in a general infinitesimal triangular array. In the case of
positive
variables, the main theorem extends a result proved earlier for
arrays with
identically distributed rows. The case of unitary variables is
considered as
well.
http://front.math.ucdavis.edu/math.OA/0612278
---------------------------------------------------------------
4800. THE SIMILARITY METRIC
Ming Li (Univ. of Waterloo and BioInformatics Solutions Inc.) and
Xin Chen (Univ. California, Santa Barbara), Xin Li (Univ. Western
Ontario), Bin
Ma (Univ. Western Ontario), Paul Vitanyi (CWI and Univ. of Amsterdam)
A new class of distances appropriate for measuring similarity relations
between sequences, say one type of similarity per distance, is
studied. We
propose a new ``normalized information distance'', based on the
noncomputable
notion of Kolmogorov complexity, and show that it is in this class
and it
minorizes every computable distance in the class (that is, it is
universal in
that it discovers all computable similarities). We demonstrate that
it is a
metric and call it the {\em similarity metric}. This theory forms the
foundation for a new practical tool. To evidence generality and
robustness we
give two distinctive applications in widely divergent areas using
standard
compression programs like gzip and GenCompress. First, we compare whole
mitochondrial genomes and infer their evolutionary history. This
results in a
first completely automatic computed whole mitochondrial phylogeny tree.
Secondly, we fully automatically compute the language tree of 52
different
languages.
http://front.math.ucdavis.edu/cs.CC/0111054
---------------------------------------------------------------
4801. A NEW QUARTET TREE HEURISTIC FOR HIERARCHICAL CLUSTERING
Rudi Cilibrasi and Paul M.B. Vitanyi
We consider the problem of constructing an an optimal-weight tree
from the
3*(n choose 4) weighted quartet topologies on n objects, where
optimality means
that the summed weight of the embedded quartet topologiesis optimal
(so it can
be the case that the optimal tree embeds all quartets as non-optimal
topologies). We present a heuristic for reconstructing the optimal-
weight tree,
and a canonical manner to derive the quartet-topology weights from a
given
distance matrix. The method repeatedly transforms a bifurcating tree,
with all
objects involved as leaves, achieving a monotonic approximation to
the exact
single globally optimal tree. This contrasts to other heuristic
search methods
from biological phylogeny, like DNAML or quartet puzzling, which,
repeatedly,
incrementally construct a solution from a random order of objects, and
subsequently add agreement values.
http://front.math.ucdavis.edu/cs.DS/0606048
---------------------------------------------------------------
4802. TWO-PLAYER KNOCK 'EM DOWN
James Allen Fill and David B. Wilson
We analyze the two-player game of Knock 'em Down, asymptotically as the
number of tokens to be knocked down becomes large. Optimal play
requires mixed
strategies with deviations of order sqrt(n) from the naive law-of-
large numbers
allocation. Upon rescaling by sqrt(n) and sending n to infinity, we
show that
optimal play's random deviations always have bounded support and have
marginal
distributions that are absolutely continuous with respect to Lebesgue
measure.
http://front.math.ucdavis.edu/math.PR/0612205
---------------------------------------------------------------
4803. SINAI'S WALK: A STATISTICAL ASPECT
Pierre Andreoletti (MAPMO)
We consider Sinai's random walk in random environment. We prove that the
logarithm of the local time is a good estimator of the random potential
associated to the random environment. We give a constructive method
allowing us
to built the random environment from a single trajectory of the
random walk.
http://front.math.ucdavis.edu/math.PR/0612209
---------------------------------------------------------------
4804. A FILTERING APPROACH TO TRACKING VOLATILITY FROM PRICES
OBSERVED AT RANDOM TIMES
Jak\v{s}a Cvitani\'{c} and Robert Liptser and Boris Rozovskii
This paper is concerned with nonlinear filtering of the coefficients
in asset
price models with stochastic volatility. More specifically, we assume
that the
asset price process $S=(S_{t})_{t\geq0}$ is given by \[
dS_{t}=m(\theta_{t})S_{t} dt+v(\theta_{t})S_{t} dB_{t}, \] where
$B=(B_{t})_{t\geq0}$ is a Brownian motion, $v$ is a positive function
and
$\theta=(\theta_{t})_{t\geq0}$ is a c\'{a}dl\'{a}g strong Markov
process. The
random process $\theta$ is unobservable. We assume also that the
asset price
$S_{t}$ is observed only at random times $0<\tau_{1}<\tau_{2}<....$
This is an
appropriate assumption when modeling high frequency financial data
(e.g.,
tick-by-tick stock prices).
In the above setting the problem of estimation of $\theta$ can be
approached
as a special nonlinear filtering problem with measurements generated
by a
multivariate point process $(\tau_{k},\log S_{\tau_{k}})$. While
quite natural,
this problem does not fit into the ``standard'' diffusion or simple
point
process filtering frameworks and requires more technical tools. We
derive a
closed form optimal recursive Bayesian filter for $\theta_{t}$, based
on the
observations of $(\tau_{k},\log S_{\tau_{k}})_{k\geq1}$. It turns out
that the
filter is given by a recursive system that involves only deterministic
Kolmogorov-type equations, which should make the numerical
implementation
relatively easy.
http://front.math.ucdavis.edu/math.PR/0612212
---------------------------------------------------------------
4805. A SLOW TRANSIENT DIFFUSION IN A DRIFTED STABLE POTENTIAL
Arvind Singh (PMA)
We consider a diffusion process $X$ in a random potential $\V$ of the
form
$\V_x = \S_x -\delta x$ where $\delta$ is a positive drift and $\S$ is a
strictly stable process of index $\alpha\in (1,2)$ with positive
jumps. Then
the diffusion is transient and $X_t / \log^\alpha t$ converges in law
towards
an exponential distribution. This behaviour contrasts with the case
where $\V$
is a drifted Brownian motion and provides an example of a transient
diffusion
in a random potential which is as "slow" as in the recurrent setting.
http://front.math.ucdavis.edu/math.PR/0612220
---------------------------------------------------------------
4806. DUALITY AND EXACT CORRELATIONS FOR A MODEL OF HEAT CONDUCTION
C. Giardin\'a and J. Kurchan and F. Redig
We study a model of heat conduction with stochastic diffusion of
energy. We
obtain a dual particle process which describes the evolution of all the
correlation functions. An exact expression for the covariance of the
energy
exhibits long-range correlations in the presence of a current. We
discuss the
formal connection of this model with the simple symmetric exclusion
process.
http://front.math.ucdavis.edu/cond-mat/0612198
---------------------------------------------------------------
4807. BESSEL POTENTIALS, HITTING DISTRIBUTIONS AND GREEN FUNCTIONS
T. Byczkowski and M. Ryznar and J. Malecki
The purpose of this paper is to find explicit formulas for basic objects
pertaining the local potential theory of the operator $(I-\Delta)^
{\alpha/2}$,
$0<\alpha<2$. The potential theory of this operator is based on Bessel
potentials $J_{\alpha}=(I-\Delta)^{-\alpha/2}$. We compute the {\it
harmonic
measure} of the half-space and write a concise form of the
corresponding {\it
Green function} for the operator $(I-\Delta)^{\alpha/2}$. To achieve
this we
analyze the so-called {\it relativistic $\alpha$-stable process} on $
\R^d$
space, killed when exiting the half-space. In terms of this process
we are
dealing here with the 1-{\it potential theory} or, equivalently,
potential
theory of Schr{\"o}dinger operator based on the generator of the
process with
Kato's potential $q=-1$.
http://front.math.ucdavis.edu/math.PR/0612176
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4808. THE LIMITING SPECTRA OF GIRKO'S BLOCK-MATRIX
Tamer Oraby
To analyze the limiting spectral distribution of some random block-
matrices,
Girko [Girko, 2000] uses a system of canonical equations from [Girko,
98]. In
this paper, we use the method of moments to give an integral form for
the
almost sure limiting spectral distribution of such matrices.
http://front.math.ucdavis.edu/math.PR/0612177
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4809. UTILITY MAXIMIZATION IN A JUMP MARKET MODEL
Marie-Amelie Morlais
In this paper, we consider the classical problem of utility
maximization in a
financial market allowing jumps. Assuming that the constraint set is
a compact
set, rather than a convex one, we use a dynamic method from which we
derive a
specific BSDE. We then aim at showing existence and uniqueness
results for the
introduced BSDE. This allows us to give an explicit expression of the
value
function and characterize optimal strategies for our problem.
http://front.math.ucdavis.edu/math.PR/0612181
---------------------------------------------------------------
4810. AN ARITHMETIC MODEL FOR THE TOTAL DISORDER PROCESS
C. P. Hughes and A. Nikeghbali and M. Yor
We prove a multidimensional extension of Selberg's central limit
theorem for
the logarithm of the Riemann zeta function on the critical line. The
limit is a
totally disordered process, whose coordinates are all independent and
Gaussian.
http://front.math.ucdavis.edu/math.PR/0612195
---------------------------------------------------------------
4811. BROWNIAN SUPER-EXPONENTS
Victor Goodman (Indiana University)
We introduce a transform on the class of stochastic exponentials for
d-dimensional Brownian motions. Each stochastic exponential generates
another
stochastic exponential under the transform. The new exponential
process is
often merely a supermartingale even in cases where the original
process is a
martingale. We determine a necessary and sufficient condition for the
transform
to be a martingale process. The condition links expected values of the
transformed stochastic exponential to the distribution function of
certain
time-integrals.
http://front.math.ucdavis.edu/math.PR/0612160
---------------------------------------------------------------
4812. APPROCHE INTRINS\`{E}QUE DES FLUCTUATIONS QUANTIQUES EN M\'{E}
CANIQUE STOCHASTIQUE (AN INTRINSIC APPROACH OF THE QUANTUM
FLUCTUATIONS IN STOCHASTIC
MECHANICS)
Michel Fliess (INRIA Futurs)
This note is answering an old questioning about the F\'{e}nyes-Nelson
stochastic mechanics. The Brownian nature of the quantum
fluctuations, which
are associated to this mechanics, is deduced from Feynman's
interpretation of
the Heisenberg uncertainty principle via infinitesimal random walks
stemming
from nonstandard analysis. It is therefore no more necessary to
combine those
fluctuations with a background field, which has never been well
understood.
Most of the technical details are contained in an extended english
abstract.
http://front.math.ucdavis.edu/quant-ph/0612033
---------------------------------------------------------------
4813. MARKOV LOOPS, DETERMINANTS AND GAUSSIAN FIELDS
Yves Le Jan (LM-Orsay)
The purpose of this note is to explore some simple relations between
loop
measures, determinants, and Gaussian Markov fields.
http://front.math.ucdavis.edu/math.PR/0612112
---------------------------------------------------------------
4814. SQUARE SUMMABILITY OF VARIATIONS AND CONVERGENCE OF THE
TRANSFER OPERATOR
Anders Johansson and Anders \"Oberg
In this paper we study the one-sided shift operator on a state space
defined
by a finite alphabet. Using a scheme developed by Walters [13], we
prove that
the sequence of iterates of the transfer operator converges under square
summability of variations of the g-function, a condition which gave
uniqueness
of a g-measure in [7]. We also prove uniqueness of so-called G-measures,
introduced by Brown and Dooley [2], under square summability of
variations.
http://front.math.ucdavis.edu/math.DS/0612131
---------------------------------------------------------------
4815. COMPUTABLE EXPONENTIAL BOUNDS FOR SCREENED ESTIMATION AND
SIMULATION
I. Kontoyiannis and S.P. Meyn
Suppose the expectation E(F(X)) is to be estimated by the empirical
averages
of the values of F on independent and identically distributed samples
{X_i}. A
sampling rule called the ``screened'' estimator is introduced, and its
performance is studied. When the mean E(U(X)) of a different function
U is
known, the estimates are ``screened,'' in that we only consider those
which
correspond to times when the empirical average of the {U(X_i)} is
sufficiently
close to its known mean. As long as U dominates F appropriately, the
screened
estimates admit exponential error bounds, even when F(X) is heavy-
tailed. The
main results are several nonasymptotic, explicit exponential bounds
for the
screened estimates. A geometric interpretation, in the spirit of Sanov's
theorem, is given for the fact that the screened estimates always admit
exponential error bounds, even if the standard estimates do not. And
when they
do, the screened estimates' error probability has a significantly better
exponent. This implies that screening can be interpreted as a variance
reduction technique. Our main mathematical tools come from large
deviations
techniques. The results are illustrated by a detailed simulation
example.
http://front.math.ucdavis.edu/math.PR/0612040
---------------------------------------------------------------
4816. ON THE SUBMODULARITY OF INFLUENCE IN SOCIAL NETWORKS
Elchanan Mossel and Sebastien Roch
We prove and extend a conjecture of Kempe, Kleinberg, and Tardos
(KKT) on the
spread of influence in social networks. A social network can be
represented by
a directed graph where the nodes are individuals and the edges
indicate a form
of social relationship. A simple way to model the diffusion of ideas,
innovative behavior, or ``word-of-mouth'' effects on such a graph is to
consider an increasing process of ``infected'' (or active) nodes:
each node
becomes infected once an activation function of the set of its infected
neighbors crosses a certain threshold value. Such a model was
introduced by KKT
in \cite{KeKlTa:03,KeKlTa:05} where the authors also impose several
natural
assumptions: the threshold values are (uniformly) random; and the
activation
functions are monotone and submodular. For an initial set of active
nodes $S$,
let $\sigma(S)$ denote the expected number of active nodes at
termination. Here
we prove a conjecture of KKT: we show that the function $\sigma(S)$ is
submodular under the assumptions above. We prove the same result for the
expected value of any monotone, submodular function of the set of
active nodes
at termination.
http://front.math.ucdavis.edu/math.PR/0612046
---------------------------------------------------------------
4817. ATTRACTION TIME FOR STRONGLY REINFORCED WALKS
C. Cotar and V. Limic
We consider a class of strongly edge reinforced random walks, where the
corresponding reinforcement weight function is non-decreasing. It is
known by
Limic and Tarr\`es (2006) that the attracting edge emerges with
probability 1,
whenever the underlying graph is locally bounded. We study the
asymptotic
behavior of the tail distribution of the (random) time of attraction. In
particular, we obtain exact (up to multiplicative constant)
asymptotics if the
underlying graph has two edges. Next we show some extensions in the
setting of
finite and bounded degree infinite graphs. A nice corollary is that
if the
reinforcement weight has the form $W(k) = k^\rho$, $\rho>1$, then
(universally
over finite graphs) the expected time to attraction is infinite if
and only if
$\rho \leq 1+ \frac{1+\sqrt{5}}{2}$.
http://front.math.ucdavis.edu/math.PR/0612048
---------------------------------------------------------------
4818. CONVERGENCE OF SEQUENTIAL MARKOV CHAIN MONTE CARLO METHODS: I.
NONLINEAR FLOW OF PROBABILITY MEASURES
Andreas Eberle and Carlo Marinelli
Sequential Monte Carlo Samplers are a class of stochastic algorithms for
Monte Carlo integral estimation w.r.t. probability distributions,
which combine
elements of Markov chain Monte Carlo methods and importance sampling/
resampling
schemes. We develop a stability analysis by functional inequalities
for a
nonlinear flow of probability measures describing the limit behavior
of the
algorithms as the number of particles tends to infinity. Stability
results are
derived both under global and local assumptions on the generator of the
underlying Metropolis dynamics. This allows us to prove that the
combined
methods sometimes have good asymptotic stability properties in
multimodal
setups where traditional MCMC methods mix extremely slowly. For
example, this
holds for the mean field Ising model at all temperatures.
http://front.math.ucdavis.edu/math.PR/0612074
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4819. OPTION PRICING WITHOUT PRICE DYNAMICS: A PROBABILISTIC APPROACH
Dimitris Bertsimas and Natasha Bushueva
Employing probabilistic techniques we compute best possible upper and
lower
bounds on the price of an option on one or two assets with continuous
piecewise
linear payoff function based on prices of simple call options of
possibly
distinct maturities and the no-arbitrage condition, but without any
assumption
on the price dynamics of underlying assets. We show that the problem
reduces to
solving linear optimization problems that we explicitly characterize.
We report
numerical results that illustrate the effectiveness of the algorithms we
develop.
http://front.math.ucdavis.edu/math.PR/0612075
---------------------------------------------------------------
4820. A SINGULAR PERTURBATION APPROACH FOR CHOOSING PAGERANK DAMPING
FACTOR
Konstantin Avrachenkov and Nelly Litvak and Kim Son Pham
The choice of the PageRank damping factor is not evident. The
Google's choice
for the value c=0.85 was a compromise between the true reflection of
the Web
structure and numerical efficiency. However, the Markov random walk
on the
original Web Graph does not reflect the importance of the pages
because it
absorbs in dead ends. Thus, the damping factor is needed not only for
speeding
up the computations but also for establishing a fair ranking of
pages. In this
paper, we propose new criteria for choosing the damping factor, based
on the
ergodic structure of the Web Graph and probability flows.
Specifically, we
require that the core component receives a fair share of the PageRank
mass.
Using singular perturbation approach we conclude that the value
c=0.85 is too
high and suggest that the damping factor should be chosen around 1/2.
As a
by-product, we describe the ergodic structure of the OUT component of
the Web
Graph in detail. Our analytical results are confirmed by experiments
on two
large samples of the Web Graph.
http://front.math.ucdavis.edu/math.PR/0612079
---------------------------------------------------------------
4821. HYDRODYNAMICS AND HYDROSTATICS FOR A CLASS OF ASYMMETRIC
PARTICLE SYSTEMS WITH OPEN BOUNDARIES
Christophe Bahadoran
We consider asymmetric attractive particle systems with product
invariant
measures in any space dimension. We show that, in the presence of open
boundaries, the hydrodynamic limit is a scalar conservation law with
boundary
conditions in the sense defined by Bardos, Leroux and N\'{e}d\'{e}
lec. When the
boundaries are parallel hyperplanes, we establish a large-time
convergence
result for the entropy solution and derive the stationary profile for
the
particle system. Models include current-density relations with
arbitrarily many
maxima and minima.
http://front.math.ucdavis.edu/math.PR/0612094
---------------------------------------------------------------
4822. THE PAVING PROPERTY FOR UNIFORMLY BOUNDED MATRICES: A NEW PROOF
Joel A. Tropp
This note presents a new proof of an important result due to Bourgain
and
Tzafriri that provides a partial solution to the Kadison--Singer
problem. The
result shows that every unit-norm matrix whose entries are relatively
small in
comparison with its dimension can be paved by a partition of constant
size.
That is, the coordinates can be partitioned into a constant number of
blocks so
that the restriction of the matrix to each block of coordinates has
norm less
than one half. The original proof of Bourgain and Tzafriri involves a
long,
delicate calculation. The new proof relies on the systematic use of
symmetrization and Khintchine inequalities to estimate the norm of
some random
matrices.
http://front.math.ucdavis.edu/math.MG/0612070
---------------------------------------------------------------
4823. RELATIVISTIC DIFFUSION IN G\"ODEL'S UNIVERSE
Jacques Franchi
K. G\"odel [G] discovered his celebrated solution to Einstein
equations in
1949. Additional contributions were made by Kundt [K] and Hawking-Ellis
([H-E],5.7). On the other hand, a general Lorentz invariant operator,
associated to the so-called "relativistic diffusion'', and making
sense in any
Lorentz manifold, was introduced by Franchi-Le Jan in [F-LJ]. Here is
purposed
a first study of the relativistic diffusion in the framework of G
\"odel's
universe, which contains matter.
http://front.math.ucdavis.edu/math.PR/0612020
---------------------------------------------------------------
4824. MODIFIED LOGARITHMIC SOBOLEV INEQUALITIES ON R
Franck Barthe (LSProba) and Cyril Roberto (LAMA)
We provide a sufficient condition for a measure on the real line to
satisfy a
modified logarithmic Sobolev inequality, thus extending the criterion
of Bobkov
and G\"{o}tze. Under mild assumptions the condition is also necessary.
Concentration inequalities are derived. This completes the picture
given in
recent contributions by Gentil, Guillin and Miclo.
http://front.math.ucdavis.edu/math.PR/0612026
---------------------------------------------------------------
4825. EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION
Victor Goodman and Kyounghee Kim
We find a simple expression for the probability density of $\int \exp
(B_s -
s/2) ds$ in terms of its distribution function and the distribution
function
for the time integral of $\exp (B_s + s/2)$. The relation is obtained
with a
change of measure argument where expectations over events determined
by the
time integral are replaced by expectations over the entire
probability space.
We develop precise information concerning the lower tail
probabilities for
these random variables as well as for time integrals of geometric
Brownian
motion with arbitrary constant drift. In particular, $E[ \exp\big
(\theta / \int
\exp (B_s)ds\big) ]$ is finite iff $\theta < 2$. We present a new
formula for
the price of an Asian call option.
http://front.math.ucdavis.edu/math.PR/0612034
---------------------------------------------------------------
4826. ONE-FACTOR TERM STRUCTURE WITHOUT FORWARD RATES
Victor Goodman and Kyounghee Kim
We construct a no-arbitrage model of bond prices where the long bond
is used
as a numeraire. We develop bond prices and their dynamics without
developing
any model for the spot rate or forward rates. The model is arbitrage
free and
all nominal interest rates remain positive in the model. We give
examples where
our model does not have a spot rate; other examples include both spot
and
forward rates.
http://front.math.ucdavis.edu/math.PR/0612035
---------------------------------------------------------------
4827. CONFORMAL BOUNDARY LOOP MODELS
Jesper Lykke Jacobsen (LPTMS and SPhT) and Hubert Saleur (SPhT)
We study a model of densely packed self-avoiding loops on the annulus,
related to the Temperley Lieb algebra with an extra idempotent boundary
generator. Four different weights are given to the loops, depending
on their
homotopy class and whether they touch the outer rim of the annulus.
When the
weight of a contractible bulk loop x = q + 1/q satisfies -2 < x <= 2,
this
model is conformally invariant for any real weight of the remaining
three
parameters. We classify the conformal boundary conditions and give exact
expressions for the corresponding boundary scaling dimensions. The
amplitudes
with which the sectors with any prescribed number and types of non
contractible
loops appear in the full partition function Z are computed
rigorously. Based on
this, we write a number of identities involving Z which hold true for
any
finite size. When the weight of a contractible boundary loop y takes
certain
discrete values, y_r = [r+1]_q / [r]_q with r integer, other identities
involving the standard characters K_{r,s} of the Virasoro algebra are
established. The connection with Dirichlet and Neumann boundary
conditions in
the O(n) model is discussed in detail, and new scaling dimensions are
derived.
When q is a root of unity and y = y_r, exact connections with the A_m
type RSOS
model are made. These involve precise relations between the spectra
of the loop
and RSOS model transfer matrices, valid in finite size. Finally, the
results
where y=y_r are related to the theory of Temperley Lieb cabling.
http://front.math.ucdavis.edu/math-ph/0611078
---------------------------------------------------------------
4828. MARKOV CHAIN APPROXIMATIONS FOR SYMMETRIC JUMP PROCESSES
R. Husseini and M. Kassmann
Markov chain approximations of symmetric jump processes are
investigated.
Tightness results and a central limit theorem are established.
Moreover, given
the generator of a symmetric jump process with state space $\mathbbm
{R}^d$ the
approximating Markov chains are constructed explicitly. As a
byproduct we
obtain a definition of the Sobolev space $H^{\alpha/2}(\mathbbm{R}^d)
$, $\alpha
\in (0,2)$, that is equivalent to the standard one.
http://front.math.ucdavis.edu/math.PR/0611934
---------------------------------------------------------------
4829. DOES WASTE-RECYCLING REALLY IMPROVE METROPOLIS-HASTINGS MONTE
CARLO ALGORITHM?
Jean-Fran\c{c}ois Delmas (CERMICS) and Benjamin Jourdain (CERMICS)
The waste-recycling Monte Carlo (WR) algorithm, introduced by
Frenkel, is a
modification of the Metropolis-Hastings algorithm, which makes use of
all the
proposals, whereas the standard Metropolis-Hastings algorithm only
uses the
accepted proposals. We prove the convergence of the WR algorithm and its
asymptotic normality. We give an example which shows that in general
the WR
algorithm is not asymptotically better than the Metropolis-Hastings
algorithm :
the WR algorithm can have an asymptotic variance larger than the one
of the
Metropolis-Hastings algorithm. However, in the particular case of the
Metropolis-Hastings algorithm called Boltzmann algorithm, we prove
that the WR
algorithm is asymptotically better than the Metropolis-Hastings
algorithm.
http://front.math.ucdavis.edu/math.PR/0611949
---------------------------------------------------------------
4830. SPARSITY AND INCOHERENCE IN COMPRESSIVE SAMPLING
Emmanuel Candes and Justin Romberg
We consider the problem of reconstructing a sparse signal $x^0\in\R^n
$ from a
limited number of linear measurements. Given $m$ randomly selected
samples of
$U x^0$, where $U$ is an orthonormal matrix, we show that $\ell_1$
minimization
recovers $x^0$ exactly when the number of measurements exceeds \[ m\geq
\mathrm{Const}\cdot\mu^2(U)\cdot S\cdot\log n, \] where $S$ is the
number of
nonzero components in $x^0$, and $\mu$ is the largest entry in $U$
properly
normalized: $\mu(U) = \sqrt{n} \cdot \max_{k,j} |U_{k,j}|$. The
smaller $\mu$,
the fewer samples needed.
The result holds for ``most'' sparse signals $x^0$ supported on a
fixed (but
arbitrary) set $T$. Given $T$, if the sign of $x^0$ for each nonzero
entry on
$T$ and the observed values of $Ux^0$ are drawn at random, the signal is
recovered with overwhelming probability. Moreover, there is a sense
in which
this is nearly optimal since any method succeeding with the same
probability
would require just about this many samples.
http://front.math.ucdavis.edu/math.ST/0611957
---------------------------------------------------------------
4831. GRAVITATIONAL ALLOCATION TO POISSON POINTS
Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik
For d>=3, we construct a non-randomized, fair and translation-
equivariant
allocation of Lebesgue measure to the points of a standard Poisson point
process in R^d, defined by allocating to each of the Poisson points
its basin
of attraction with respect to the flow induced by a gravitational
force field
exerted by the points of the Poisson process. We prove that this
allocation
rule is economical in the sense that the "allocation diameter",
defined as the
diameter X of the basin of attraction containing the origin, is a random
variable with a rapidly decaying tail. Specifically, we have the tail
bound:
P(X > R) < C exp[ -c R(log R)^(alpha_d) ], for all R>2, where: alpha_d =
(d-2)/d for d>=4; alpha_3 can be taken as any number <-4/3; and C,c are
positive constants that depend on d and alpha_d. This is the first
construction
of an allocation rule of Lebesgue measure to a Poisson point process
with
subpolynomial decay of the tail P(X>R).
http://front.math.ucdavis.edu/math.PR/0611886
---------------------------------------------------------------
4832. AN EXTENSION OF THE LEVY CHARACTERIZATION TO FRACTIONAL
BROWNIAN MOTION
Yulia Mishura and Esko Valkeila
We extend the classical Levy characterization of Brownian motion to
fractional Brownian motion.
http://front.math.ucdavis.edu/math.PR/0611913
---------------------------------------------------------------
4833. A FUNCTIONAL NON-CENTRAL LIMIT THEOREM FOR JUMP-DIFFUSIONS WITH
PERIODIC COEFFICIENTS DRIVEN BY STABLE LEVY-NOISE
Brice Franke
We prove a functional non-central limit theorem for jump-diffusions with
periodic coefficients driven by strictly stable Levy-processes with
stability
index bigger than one. The limit process turns out to be a strictly
stable Levy
process with an averaged jump-measure. Unlike in the situation where the
diffusion is driven by Brownian motion, there is no drift related
enhancement
of diffusivity.
http://front.math.ucdavis.edu/math.PR/0611852
---------------------------------------------------------------
4834. HOW DO RANDOM FIBONACCI SEQUENCES GROW?
Elise Janvresse (LMRS) and Beno\^{i}t Rittaud (IG) and Thierry De
La Rue (LMRS)
We study two kinds of random Fibonacci sequences defined by $F_1=F_2=1
$ and
for $n\ge 1$, $F_{n+2} = F_{n+1} \pm F_{n}$ (linear case) or $F_{n+2} =
|F_{n+1} \pm F_{n}|$ (non-linear case), where each sign is
independent and
either + with probability $p$ or - with probability $1-p$ ($0<p\le 1
$). Our
main result is that the exponential growth of $F_n$ for $0<p\le 1$
(linear
case) or for $1/3\le p\le 1$ (non-linear case) is almost surely given by
$$\int_0^\infty \log x d\nu_\alpha (x), $$ where $\alpha$ is an explicit
function of $p$ depending on the case we consider, and $\nu_\alpha$
is an
explicit probability distribution on $\RR_+$ defined inductively on
Stern-Brocot intervals. In the non-linear case, the largest Lyapunov
exponent
is not an analytic function of $p$, since we prove that it is equal
to zero for
$0<p\le1/3$. We also give some results about the variations of the
largest
Lyapunov exponent, and provide a formula for its derivative.
http://front.math.ucdavis.edu/math.PR/0611860
---------------------------------------------------------------
4835. LAGUERRE PROCESS AND GENERALISED HARTMAN-WATSON LAW
Nizar Demni (PMA)
In this paper, we study complex Wishart processes or the so-called
Laguerre
processes. We give some interest to the behaviour of the eigenvalues
process,
derive some useful SDE and compute both infinitesimal generator and
semi-group.
We also give absolute-continuity relations between different
indices.Then, we
compute the density function of the generalised Hartman-Watson law as
well as
the law of the first hitting time of 0 when the size m=2.
http://front.math.ucdavis.edu/math.PR/0611863
---------------------------------------------------------------
4836. THE NUMERICAL ALGORITHMS AND SIMULATIONS FOR BSDES
Shige Peng and Mingyu Xu
Here we study a new numerical method for BSDE. Then we present a
package of
our numerical algorithms of BSDE with convenient user--machine
interface. This
package permit us to solve BSDE, reflected BSDE with one or two
barriers as
well as BSDE with constraints. One of significant advantages of this
package is
that users have a very convenient interface. Any users who know the
ABC of BSDE
can use this package very easily. The interface of the input-output
was also
carefully designed.
http://front.math.ucdavis.edu/math.PR/0611864
---------------------------------------------------------------
4837. ASYMPTOTIC HOMOLOGY OF THE QUOTIENT OF $PSL_2(\BR)$ BY A
MODULAR GROUP
Jacques Franchi
Consider $ G:= PSL_2(\R)\equiv T^1\H^2$, a modular group $ \Gamma$,
and the
homogeneous space $ \Gamma\sm G \equiv T^1(\Gamma\sm\H^2)$. Endow $ G
$, and
then $ \Gamma\sm G $, with a canonical left-invariant metric, thereby
equipping
it with a quasi hyperbolic geometry. Windings around handles and
cusps of $
\Gamma\sm G $ are calculated by integrals of closed 1-forms of $
\Gamma\sm G $.
The main results express, in both Brownian and geodesic cases, the joint
convergence of the law of these integrals, with a stress on the
asymptotic
independence between slow and fast windings. The non-hyperbolicity of $
\Gamma\sm G $ is responsible for a difference between the Brownian
and geodesic
asymptotic behaviours, difference which does not exist at the level
of the
Riemann surface $\Gamma\sm\H^2$ (and generally in hyperbolic cases).
Identification of the cohomology classes of closed 1-forms and with
harmonic
1-forms, and equidistribution of large geodesic spheres, are also
addressed.
http://front.math.ucdavis.edu/math.PR/0611866
---------------------------------------------------------------
4838. CORRELATION LENGTHS FOR RANDOM POLYMER MODELS AND FOR SOME
RENEWAL SEQUENCES
F. L. Toninelli (ENS Lyon and Umr--CNRS 5672)
We consider models of directed polymers interacting with a one-
dimensional
defect line on which random charges are placed. More abstractly, one
starts
from renewal sequence on Z and gives a random (site-dependent) reward or
penalty to the occurrence of a renewal at any given point of Z. These
models
are known to undergo a delocalization-localization transition, and
the free
energy $\tf$ vanishes when the critical point is approached from the
localized
region. We prove that the quenched correlation length $\xi$, defined
as the
inverse of the rate of exponential decay of the two-point function,
does not
diverge faster than $ 1/F$. We prove a lower bound also for the rate of
exponential decay of the disorder-averaged two-point function. We
discuss how,
in the particular case where disorder is absent, this result can be
seen as a
refinement of the classical renewal theorem, for a specific class of
renewal
sequences.
http://front.math.ucdavis.edu/math.PR/0611868
---------------------------------------------------------------
4839. REFLECTED BSDE WITH A CONSTRAINT AND A NEW DOOB-MEYER
NONLINEAR DECOMPOSITION
Shige Peng and Mingyu Xu
In this paper, we study a type of reflected BSDE with a constraint and
introduce a new kind of nonlinear expectation via BSDE with a
constraint and
prove the Doob-Meyer decomposition with respect to the super(sub)
martingale
introduced by this nonlinear expectation. We then apply the results
to the
pricing of American options in incomplete market.
http://front.math.ucdavis.edu/math.PR/0611869
---------------------------------------------------------------
4840. REFLECTED BSDE WITH MONOTONICITY AND GENERAL INCREASING IN $Y$,
AND NON-LIPSCHITZ CONDITIONS IN $Z$
Mingyu Xu
In this paper, we study the reflected BSDE with one continuous
barrier, under
the monotonicity and general increasing condition on $y$ and non
Lipschitz
condition on $z$. We prove the existence and uniqueness of the
solution to
these equation by approximation method.
http://front.math.ucdavis.edu/math.PR/0611870
---------------------------------------------------------------
4841. A GENERALIZATION AND EXTENSION OF AN AUTOREGRESSIVE MODEL
S Satheesh and E Sandhya and K E Rajasekharan
Generalizations and extensions of a first order autoregressive model of
Lawrance and Lewis (1981) are considered and characterized here.
http://front.math.ucdavis.edu/math.PR/0611878
---------------------------------------------------------------
4842. A LAW OF LARGE NUMBERS FOR RANDOM PARTITIONS OF THE INTERVAL
AND THE LIMITING SEARCH-COST OF THE MOVE-TO-FRONT STRATEGY
Javiera Barrera and Joaquin Fontbona
We prove a law of large numbers for certain finite random partitions of
$[0,1]$, when the number of fragments go to $\infty$. Then, we apply
it to
compute the limiting distribution of the transient search-cost of the
move-to-front rule for general classes of random and deterministic
request
probabilities, when the list size goes to $\infty$.
http://front.math.ucdavis.edu/math.PR/0611882
---------------------------------------------------------------
4843. LARGE DEVIATIONS FOR STATISTICS OF JACOBI PROCESS
Nizar Demni (PMA) and Marguerite Zani (LAMA)
In this paper, we derive a handable expression for the Jacobi process
semi
group which is given by a bilinear series involving Jacobi
polynomials. Our
attempt uses a subordination of the considered process by means of a
suitable
random change. Once we did, we will be able, in the ultraspheric
case, to
derive a LDP for a family of estimators based on a single trajectory
of the
process.
http://front.math.ucdavis.edu/math.PR/0611884
---------------------------------------------------------------
4844. PROBING RARE PHYSICAL TRAJECTORIES WITH LYAPUNOV WEIGHTED DYNAMICS
Julien Tailleur and Jorge Kurchan
The transition from order to chaos has been a major subject of
research since
the work of Poincare, as it is relevant in areas ranging from the
foundations
of statistical physics to the stability of the solar system. Along this
transition, atypical structures like the first chaotic regions to
appear, or
the last regular islands to survive, play a crucial role in many
physical
situations. For instance, resonances and separatrices determine the
fate of
planetary systems, and localised objects like solitons and breathers
provide
mechanisms of energy transport in nonlinear systems such as Bose-
Einstein
condensates and biological molecules. Unfortunately, despite the
fundamental
progress made in the last years, most of the numerical methods to
locate these
'rare' trajectories are confined to low-dimensional or toy models,
while the
realms of statistical physics, chemical reactions, or astronomy are
still hard
to reach. Here we implement an efficient method that allows one to
work in
higher dimensions by selecting trajectories with unusual chaoticity.
As an
example, we study the Fermi-Pasta-Ulam nonlinear chain in equilibrium
and show
that the algorithm rapidly singles out the soliton solutions when
searching for
trajectories with low level of chaoticity, and chaotic-breathers in the
opposite situation. We expect the scheme to have natural applications in
celestial mechanics and turbulence, where it can readily be combined
with
existing numerical methods
http://front.math.ucdavis.edu/cond-mat/0611672
---------------------------------------------------------------
4845. A NOTE ON TALAGRAND'S CONVEX HULL CONCENTRATION INEQUALITY
David Pollard
The paper reexamines an argument by Talagrand that leads to a remarkable
exponential tail bound for the concentration of probability near a
set. The
main novelty is the replacement of a mysterious Calculus inequality
by an
application of Jensen's inequality.
http://front.math.ucdavis.edu/math.PR/0611770
---------------------------------------------------------------
4846. COLORED LOOP-ERASED RANDOM WALK ON THE COMPLETE GRAPH
Jomy Alappattu and Jim Pitman
Starting from a sequence regarded as a walk through some set of
values, we
consider the associated loop-erased walk as a sequence of directed
edges, with
an edge from $i$ to $j$ if the loop erased walk makes a step from $i$
to $j$.
We introduce a coloring of these edges by painting edges with a fixed
color as
long as the walk does not loop back on itself, then switching to a
new color
whenever a loop is erased, with each new color distinct from all
previous
colors. The pattern of colors along the edges of the loop-erased walk
then
displays stretches of consecutive steps of the walk left untouched by
the
loop-erasure process. Assuming that the underlying sequence
generating the
loop-erased walk is a sequence of independent random variables, each
uniform on
$[N]:=\{1, 2, ..., N\}$, we condition the walk to start at $N$ and
stop the
walk when it first reaches the subset $[k]$, for some $1 \leq k \leq
N-1$. We
relate the distribution of the random length of this loop-erased walk
to the
distribution of the length of the first loop of the walk, via Cayley's
enumerations of trees, and via Wilson's algorithm. For fixed $N$ and
$k$, and
$i = 1,2, ...$, let $B_i$ denote the event that the loop-erased walk
from $N$
to $[k]$ has $i +1$ or more edges, and the $i^{th}$ and $(i+1)^{th}$
of these
edges are colored differently. We show that given that the loop-
erased random
walk has $j$ edges for some $1\leq j \leq N-k$, the events $B_i$ for
$1 \leq i
\leq j-1$ are independent, with the probability of $B_i$ equal to $1/
(k+i+1)$.
This determines the distribution of the sequence of random lengths of
differently colored segments of the loop-erased walk, and yields
asymptotic
descriptions of these random lengths as $N \to \infty$.
http://front.math.ucdavis.edu/math.PR/0611775
---------------------------------------------------------------
4847. THE COX THEOREM: UNKNOWNS AND PLAUSIBLE VALUE
Maurice J. Dupre and Frank J. Tipler
We give a proof of Cox's Theorem on the product rule and sum rule for
conditional plausibility without assuming continuity or
differentiablity of
plausibility. Instead, we extend the notion of plausibility to apply to
unknowns giving them plausible values. Our proof is enormously
simpler than
others that have recently appeared in the literature, yet completely
rigorous.
For example, we do not need to investigate the 11 possibilities for
conditional
plausibilities as described on page 25 of Jaynes' recent book
Probability
Theory.
http://front.math.ucdavis.edu/math.PR/0611795
---------------------------------------------------------------
4848. CONVOLUTION-TYPE STOCHASTIC VOLTERRA EQUATIONS WITH ADDITIVE
FRACTIONAL BROWNIAN MOTION IN HILBERT SPACE
Peter Caithamer and Anna Karczewska
We consider convolution-type stochastic Volterra equations with additive
Hilbert-valued fractional Brownian motion, $0<H<1$. We find the weak
solution
to this stochastic Volterra equation, and study its stochastic
integral part,
the stochastic convolution, which we show to be mean-zero Gaussian.
We develop
an It\^o isometry for stochastic integrals with respect to a Hilbert-
valued
fractional Brownian motion, and use it to compute the covariance of the
stochastic convolution. This formula, which uses fractional integrals
and
derivatives, generalizes the well-known formula from the case $H=1/2$.
http://front.math.ucdavis.edu/math.PR/0611832
---------------------------------------------------------------
4849. EQUILIBRIUM FLUCTUATIONS FOR THE ZERO-RANGE PROCESS ON THE
SIERPINSKI GASKET
M. D. Jara
We obtain the equilibrium fluctuations for the empirical density of
particles
for the zero-range process in the Sierpinski gasket. The limiting
process is a
generalized Ornstein-Uhlenbeck process generated by the Neumann
Laplacian and
its corresponding Dirichlet form on the gasket.
http://front.math.ucdavis.edu/math.PR/0611836
---------------------------------------------------------------
4850. PARAMETRIC ESTIMATION FOR PARTIALLY HIDDEN DIFFUSION PROCESSES
SAMPLED AT DISCRETE TIMES
Stefano Iacus and Masayuki Uchida and Nakahiro Yoshida
A one dimensional diffusion process $X=\{X_t, 0\leq t \leq T\}$ is
observed
only when its path lies over some threshold $\tau$. On the basis of the
observable part of the trajectory, the problem is to estimate finite
dimensional parameter in both drift and diffusion coefficient under a
discrete
sampling scheme. It is assumed that the sampling occurs at regularly
spaced
times intervals of length $h_n$ such that $h_n\cdot n =T$. The
asymptotic is
considered as $T\to\infty$, $n\to\infty$, $n h_n^2\to 0$. Consistency
and
asymptotic normality for estimators of parameters in both drift and
diffusion
coefficient is proved.
http://front.math.ucdavis.edu/math.ST/0611781
---------------------------------------------------------------
4851. DOBRUSHIN STATES IN THE \PHI^4_1 MODEL
L. Bertini and S. Brassesco and P. Butt\`a
We consider the van der Waals free energy functional in a bounded
interval
with inhomogeneous Dirichlet boundary conditions imposing the two
stable phases
at the endpoints. We compute the asymptotic free energy cost, as the
length of
the interval diverges, of shifting the interface from the midpoint.
We then
discuss the effect of thermal fluctuations by analyzing the \phi^4_1-
measure
with Dobrushin boundary conditions. In particular, we obtain a
nontrivial limit
in a suitable scaling in which the length of the interval diverges
and the
temperature vanishes. The limiting state is not translation invariant
and
describes a localized interface. This result can be seen as the
probabilistic
counterpart of the variational convergence of the associated excess free
energy.
http://front.math.ucdavis.edu/math-ph/0611077
---------------------------------------------------------------
4852. ECOLOGICAL EQUILIBRIUM FOR RESTRAINED BRANCHING RANDOM WALKS
Daniela Bertacchi and Gustavo Posta and Fabio Zucca
We study a generalized branching random walk where particles breed at
a rate
which depends on the number of neighbouring particles. Under general
assumptions on the breeding rates we prove the existence of a phase
where the
population survives without exploding. We construct a non trivial
invariant
measure for this case.
http://front.math.ucdavis.edu/math.PR/0611720
---------------------------------------------------------------
4853. A LATTICE GAS MODEL FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATION
J. Beltr\'an and C. Landim
We recover the Navier-Stokes equation as the incompressible limit of a
stochastic lattice gas in which particles are allowed to jump over a
mesoscopic
scale. The result holds in any dimension assuming the existence of a
smooth
solution of the Navier-Stokes equation in a fixed time interval. The
proof does
not use non-gradient methods or the multi-scale analysis due to the
long range
jumps.
http://front.math.ucdavis.edu/math.PR/0611721
---------------------------------------------------------------
4854. SELF-INTERSECTION LOCAL TIME OF $(\ALPHA,D,\BETA)$-SUPERPROCESS
L. Mytnik and J. Villa
The existence of self-intersection local time (SILT), when the time
diagonal
is intersected, of the $(\alpha,d,\beta)$-superprocess is proved for
$d/2<\alpha $ and for a renormalized SILT when $d/(2+(1+\beta)^{-1})<
\alpha
\leq d/2$. We also establish Tanaka-like formula for SILT.
http://front.math.ucdavis.edu/math.PR/0611727
---------------------------------------------------------------
4855. A RANDOM WALK WITH COLLAPSING BONDS AND ITS SCALING LIMIT
Majid Hosseini and Krishnamurthi Ravishankar
We introduce a new self-interacting random walk on the integers in a
dynamic
random environment and show that it converges to a pure diffusion in the
scaling limit. We also find a lower bound on the diffusion
coefficient in some
special cases. With minor changes the same argument can be used to
prove the
scaling limit of the corresponding walk in Z^d.
http://front.math.ucdavis.edu/math.PR/0611734
---------------------------------------------------------------
4856. AN ANALOGUE OF THE KUBILIUS MAIN THEOREM FOR QUASI-LOGARITHMIC
STRUCTURES
Bruno Nietlispach
We prove an analogue of the Kubilius main theorem for quasi-logarithmic
structures. This result extends the corresponding theorem of Arratia,
Barbour
and Tavar\'{e} (2003) in the context of logarithmic structures, and
of Zhang
(1996) in the context of additive arithmetic semigroups. In
particular, our
theorem is valid for additive arithmetic semigroups where non-classical
``Beurling type'' prime number theorems hold true.
http://front.math.ucdavis.edu/math.PR/0611747
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4857. ONE VERSION OF THE CLARK REPRESENTATION THEOREM FOR ARRATIA FLOW
Andrey A Dorogovtsev
The article contains description of the functionals from the family of
coalescing Brownian particles. New type of the stochastic integral is
introduced and used.
http://front.math.ucdavis.edu/math.PR/0611748
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4858. SMOOTHING PROBLEM IN ANTICIPATING SCENARIO
Andrey A Dorogovtsev
This article is devoted to the stochastic anticipating equations with
the
extended stochastic integral with respect to the Gaussian processes of a
special type and its application to the smoothing problem in the case
when
noise is represented by the two jointly Gaussian Wiener processes,
which can
have not a semimartingale property with respect to the joint filtration.
http://front.math.ucdavis.edu/math.PR/0611749
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4859. ONE BROWNIAN STOCHASTIC FLOW
Andrey A Dorogovtsev
The weak limits of the measure-valued processes organized as a mass
carried
by the interacting Brownian particles are described. As a limiting
flow the
Arrattia flow is obtained.
http://front.math.ucdavis.edu/math.PR/0611750
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4860. STOCHASTIC ANTICIPATING BOUNDARY VALUE PROBLEMS
Andrey A Dorogovtsev
This article is devoted to the stochastic anticipating equations with
the
extended stochastic integral with respect to the Gaussian processes of a
special type. In the particular cases the solutions of such an
equations are
the well-known Wiener functionals after the second quantization. As an
application the stochastic Kolmogorov equation for the conditional
distributions of the diffusion process is obtained. Also we will
consider the
conditional variant of the Feynman--Kac formula. The two last
sections of the
article are devoted to the smoothing problem in the case when noise is
represented by the two jointly Gaussian Wiener processes, which can
have not a
semimartingale property with respect to the joint filtration.
http://front.math.ucdavis.edu/math.PR/0611751
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4861. STATISTICAL PHYSICS ALGORITHMS FOR TRAFFIC RECONSTRUCTION
Arnaud De La Fortelle (CAOR) and Jean-Marc Lasgouttes (INRIA
Rocquencourt), Cyril Furtlehner (INRIA Rocquencourt)
Concepts and techniques from statistical physics inspired a new
method for
traffic prediction. This method is particularly suitable in settings
where the
only information available is floating car data. We propose a system,
based on
the Ising model of statistical physics, which both reconstructs and
predicts
the traffic in real time using a message-passing algorithm.
http://front.math.ucdavis.edu/math.PR/0611757
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4862. PROPHET INEQUALITIES FOR I.I.D. RANDOM VARIABLES WITH RANDOM
ARRIVAL TIMES
Pieter C. Allaart
Suppose $X_1,X_2,...$ are i.i.d. nonnegative random variables with
finite
expectation, and for each $k$, $X_k$ is observed at the $k$-th
arrival time
$S_k$ of a Poisson process with unit rate which is independent of the
sequence
$\{X_k\}$. For $t>0$, comparisons are made between the expected maximum
$M(t):=\rE[\max_{k\geq 1} X_k \sI(S_k\leq t)]$ and the optimal
stopping value
$V(t):=\sup_{\tau\in\TT}\sE[X_\tau \sI(S_\tau\leq t)]$, where $\TT$
is the set
of all $\NN$-valued random variables $\tau$ such that $\{\tau=i\}$ is
measurable with respect to the $\sigma$-algebra generated by
$(X_1,S_1),...,(X_i,S_i)$. For instance, it is shown that $M(t)/V(t)\leq
1+\alpha_0$, where $\alpha_0\doteq 0.34149$ satisfies $\int_0^1(y-y\ln
y+\alpha_0)^{-1} dy=1$; and this bound is asymptotically sharp as $t
\to\infty$.
Another result is that $M(t)/V(t)<2-(1-e^{-t})/t$, and this bound is
asymptotically sharp as $t\downarrow 0$. Upper bounds for the difference
$M(t)-V(t)$ are also given, under the additional assumption that the
$X_k$ are
bounded.
http://front.math.ucdavis.edu/math.PR/0611664
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4863. ANOMALOUS HEAT-KERNEL DECAY FOR RANDOM WALK AMONG BOUNDED
RANDOM CONDUCTANCES
Noam Berger and Marek Biskup and Christopher E. Hoffman and Gady
Kozma
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,
driven by a field of bounded random conductances $\omega_{xy}\in[0,1]
$. The
conductance law is i.i.d. subject to the condition that the
probability of
$\omega_{xy}>0$ exceeds the percolation threshold on $\Z^d$. For
environments
in which the origin is connected to infinity by bonds with positive
conductances, we prove that the $n$-step return probability $\cmss
P_\omega^n(0,0)$ is bounded by a random constant times $n^{-d/2}$ in
$d=2,3$,
while in $d\ge5$ it is bounded by a constant times $n^{-2}$. In $d=4$
we get an
upper bound proportional to $n^{-2}\log n$. The leading-order $1/n^2$
anomalous
decay in $d\ge5$ may be achieved in suitably chosen (i.i.d.)
environments.
http://front.math.ucdavis.edu/math.PR/0611666
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4864. CRITICAL PERCOLATION OF FREE PRODUCT OF GROUPS
Iva Kozakova
In this article we study percolation on the Cayley graph of a free
product of
groups. Such a graph has a tree-like structure which allows us to
evaluate the
critical values of the phase transition, mean cluster size and the
critical
exponent in bond percolation.
http://front.math.ucdavis.edu/math.PR/0611668
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4865. GROUP INVARIANT INFERRED DISTRIBUTIONS VIA NONCOMMUTATIVE
PROBABILITY
B. Heller and M. Wang
One may consider three types of statistical inference: Bayesian,
frequentist,
and group invariance-based. The focus here is on the last method. We
consider
the Poisson and binomial distributions in detail to illustrate a group
invariance method for constructing inferred distributions on
parameter spaces
given observed results. These inferred distributions are obtained
without using
Bayes' method and in particular without using a joint distribution of
random
variable and parameter. In the Poisson and binomial cases, the final
formulas
for inferred distributions coincide with the formulas for Bayes
posteriors with
uniform priors.
http://front.math.ucdavis.edu/math.PR/0611675
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4866. RANDOM WALK ON A POLYGON
Jyotirmoy Sarkar
A particle moves among the vertices of an $(m+1)$-gon which are labeled
clockwise as $0,1,...,m$. The particle starts at 0 and thereafter at
each step
it moves to the adjacent vertex, going clockwise with a known
probability $p$,
or counterclockwise with probability $1-p$. The directions of successive
movements are independent. What is the expected number of moves
needed to visit
all vertices? This and other related questions are answered using
recursive
relations.
http://front.math.ucdavis.edu/math.PR/0611676
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4867. CONFORMAL RADII FOR CONFORMAL LOOP ENSEMBLES
Oded Schramm and Scott Sheffield and David B. Wilson
The conformal loop ensembles CLE(k), defined for k in [8/3, 8], are
random
collections of loops in a planar domain which are conjectured scaling
limits of
the O(n) loop models. We calculate the distribution of the conformal
radii of
the nested loops surrounding a deterministic point. Our results agree
with
predictions made by Cardy and Ziff and by Kenyon and Wilson for the O
(n) model.
We also compute the expectation dimension of the CLE(k) gasket, which
consists
of points not surrounded by any loop, to be 2-(8-k)(3k-8)/32k, which
agrees
with the fractal dimension given by Duplantier for the O(n) model
gasket.
http://front.math.ucdavis.edu/math.PR/0611687
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4868. ONE-DIMENSIONAL RANDOM FIELD KAC'S MODEL: WEAK LARGE
DEVIATIONS PRINCIPLE
Enza Orlandi and Pierre Picco (CPT)
We prove a quenched weak large deviations principle for the Gibbs
measures of
a Random Field Kac Model (RFKM) in one dimension. The external random
magnetic
field is given by symmetrically distributed Bernoulli random
variables. The
results are valid for values of the temperature, $\beta^{-1}$, and
magnitude,
$\theta$, of the field in the region where the free energy of the
corresponding
random Curie Weiss model has only two absolute minima $m_\beta$ and
$Tm_\beta$.
We give an explicit representation of the rate functional which is a
positive
random functional determined by two distinct contributions. One is
related to
the free energy cost ${\cal F}^*$ to undergo a phase change (the surface
tension). The ${\cal F}^*$ is the cost of one single phase change and
depends
on the temperature and magnitude of the field. The other is a bulk
contribution
due to the presence of the random magnetic field. We characterize the
minimizers of this random functional. We show that they are step
functions
taking values $m_\beta$ and $Tm_\beta$. The points of discontinuity are
described by a stationary renewal process related to the $h-$extrema
for a
bilateral Brownian motion studied by Neveu and Pitman, where $h$ in
our context
is a suitable constant depending on the temperature and on magnitude
of the
random field. As an outcome we have a complete characterization of
the typical
profiles of RFKM (the ground states) which was initiated in [14] and
extended
in [16].
http://front.math.ucdavis.edu/math.PR/0611688
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4869. EXISTENCE AND UNIQUENESS OF AN INVARIANT MEASURE FOR A CHAIN
OF OSCILLATORS IN CONTACT WITH TWO HEAT BATHS
Philippe Carmona (LMJL)
In this note we consider a chain of $N$ oscillators, whose ends are in
contact with two heat baths at different temperatures. Our main
result is the
exponential convergence to the unique invariant probability measure (the
stationary state). We use the Lyapunov's function technique of Rey-
Bellet and
coauthors with different model of heat baths, and adapt these
techniques to two
new case recently considered in the literature by respectively
Bernardin and
Olla, Lefevere and Schenkel
http://front.math.ucdavis.edu/math.PR/0611689
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4870. LARGE DEVIATIONS FOR DIRICHLET PROCESSES AND POISSON-DIRICHLET
DISTRIBUTIONS WITH TWO PARAMETERS
Shui Feng
Large deviation principles are established for the two-parameter
Poisson-Dirichlet distribution and two-parameter Dirichlet process when
parameter $\theta$ approaches infinity. The motivation for these
results is to
understand the differences in terms of large deviations between the
two-parameter models and their one-parameter counterparts. New
insight is
obtained about the role of the second parameter $\alpha$ through a
comparison
with the corresponding results for the one-parameter Poisson-Dirichlet
distribution and Dirichlet process.
http://front.math.ucdavis.edu/math.PR/0611706
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4871. BOREL THEOREMS FOR RANDOM MATRICES FROM THE CLASSICAL COMPACT
SYMMETRIC SPACES
Beno\^{i}t Collins and Michael Stolz
We study random vectors of the form $({\rm Tr}(A^{(1)}V), ..., {\rm
Tr}(A^{(r)}V))$, where $V$ is a uniformly distributed element of a
matrix
version of a classical compact symmetric space, and the $A^{(\nu)}$ are
deterministic parameter matrices. We show that for increasing matrix
sizes
these random vectors converge to a joint Gaussian limit, and compute its
covariances. This generalizes previous work of Diaconis et al. for Haar
distributed matrices from the classical compact groups. The proof uses
integration formulae, due to Collins and \'{S}niady, for polynomial
functions
on the classical compact groups.
http://front.math.ucdavis.edu/math.PR/0611708
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4872. POINCAR\'{E} AND TRANSPORTATION INEQUALITIES FOR GIBBS MEASURES
UNDER THE DOBRUSHIN UNIQUENESS CONDITION
Liming Wu
In in this paper we establish an explicit and sharp estimate of the
spectral
gap (Poincar\'{e} inequality) and the transportation inequality for
Gibbs
measures, under the Dobrushin uniqueness condition. Moreover, we give a
generalization of the Liggett's $M-\epsilon$ theorem for interacting
particle
systems.
http://front.math.ucdavis.edu/math.PR/0611635
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4873. ON A STOCHASTIC VERSION OF PROUSE MODEL IN FLUID DYNAMICS
B. Ferrario and F. Flandoli
A stochastic version of a modified Navier-Stokes equation (introduced by
Prouse) is considered in a 3-dimensional torus. We prove existence and
uniqueness of martingale solutions. A different model with the non
linearity
given by a power 5 of the velocity is analyzed; for the structure
function of
this model, some insights towards an expression similar to that
obtained by the
Kolmogorov 1941 theory of turbulence are presented.
http://front.math.ucdavis.edu/math.PR/0611637
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4874. ORLICZ-SOBOLEV INEQUALITIES FOR SUB-GAUSSIAN MEASURES AND
ERGODICITY OF MARKOV SEMI-GROUPS
Cyril Roberto (LAMA) and Boguslaw Zegarlinski
We study coercive inequalities in Orlicz spaces associated to the
probability
measures on finite and infinite dimensional spaces which tails decay
slower
than the Gaussian ones. We provide necessary and sufficient criteria
for such
inequalities to hold and discuss relations between various classes of
inequalities.
http://front.math.ucdavis.edu/math.PR/0611638
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4875. AN EXTENSION TO GAUSSIAN SEMIGROUP AND SOME APPLICATIONS
Guibao Liu
We look at the semigroup generated by a system of heat equations.
Applications to testing normality and option pricing are addressed.
http://front.math.ucdavis.edu/math.PR/0611644
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4876. METASTABILITY IN INTERACTING NONLINEAR STOCHASTIC DIFFERENTIAL
EQUATIONS
Nils Berglund (CPT) and Bastien Fernandez (CPT) and Barbara Gentz
We consider the dynamics of a periodic chain of N coupled overdamped
particles under the influence of noise. Each particle is subjected to a
bistable local potential, to a linear coupling with its nearest
neighbours, and
to an independent source of white noise. We show that as the coupling
strength
increases, the number of equilibrium points of the system changes
from 3^N to
3. While for weak coupling, the system behaves like an Ising model with
spin-flip dynamics, for strong coupling (of the order N^2), it
synchronises, in
the sense that all oscillators assume almost the same position in their
respective local potential most of the time. We derive the exponential
asymptotics for the transition times, and describe the most probable
transition
paths between synchronised states, in particular for coupling
intensities below
the synchronisation threshold. Our techniques involve a centre-manifold
analysis of the desynchronisation bifurcation, with a precise control
of the
stability of bifurcating solutions, allowing us to give a detailed
description
of the system's potential landscape, in which the metastable
behaviour is
encoded.
http://front.math.ucdavis.edu/math.PR/0611647
---------------------------------------------------------------
4877. METASTABILITY IN INTERACTING NONLINEAR STOCHASTIC DIFFERENTIAL
EQUATIONS
Nils Berglund (CPT) and Bastien Fernandez (CPT) and Barbara Gentz
We consider the dynamics of a periodic chain of N coupled overdamped
particles under the influence of noise, in the limit of large N. Each
particle
is subjected to a bistable local potential, to a linear coupling with
its
nearest neighbours, and to an independent source of white noise. For
strong
coupling (of the order N^2), the system synchronises, in the sense
that all
oscillators assume almost the same position in their respective local
potential
most of the time. In a previous paper, we showed that the transition
from
strong to weak coupling involves a sequence of symmetry-breaking
bifurcations
of the system's stationary configurations, and analysed in particular
the
behaviour for coupling intensities slightly below the synchronisation
threshold, for arbitrary N. Here we describe the behaviour for any
positive
coupling intensity \gamma of order N^2, provided the particle number
N is
sufficiently large (as a function of \gamma/N^2). In particular, we
determine
the transition time between synchronised states, as well as the shape
of the
"critical droplet", to leading order in 1/N. Our techniques involve
the control
of the exact number of periodic orbits of a near-integrable twist
map, allowing
us to give a detailed description of the system's potential
landscape, in which
the metastable behaviour is encoded.
http://front.math.ucdavis.edu/math.PR/0611648
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4878. LINEAR AND QUADRATIC FUNCTIONALS OF RANDOM HAZARD RATES: AN
ASYMPTOTIC ANALYSIS
Giovanni Peccati (LSTA) and Igor Pr\"{u}nster (UNIVERSITY of Turin)
A popular Bayesian nonparametric approach to survival analysis
consists in
modeling hazard rates as kernel mixtures driven by a completely
random measure.
In this paper we derive asymptotic results for linear and quadratic
functionals
of such random hazard rates. In particular, we prove central limit
theorems for
the cumulative hazard function and for the path-second moment and
path-variance
of the hazard rate. Our techniques are based on recently established
criteria
for the weak convergence of single and double stochastic integrals
with respect
to Poisson random measures. We illustrate our results by considering
specific
models involving kernels and random measures commonly exploited in
practice.
http://front.math.ucdavis.edu/math.PR/0611652
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4879. BOURGAIN'S ENTROPY ESTIMATES REVISITED
John T. Workman
The following is a near complete set of notes of Bourgain's 1988 paper
"Almost Sure Convergence and Bounded Entropy." Both entropy results are
treated, as is one application. The proofs here are essentially those of
Bourgain's.
http://front.math.ucdavis.edu/math.CA/0611621
---------------------------------------------------------------
4880. SHAPE OF THE GROUND STATE ENERGY DENSITY OF HILL'S EQUATION
WITH NICE GAUSSIAN POTENTIAL
Jose A. Ramirez and Brian Rider
Consider Hill's operator Q = -D^2 + q(x) in which the potential q(x)
is an
almost surely continuous and rotation invariant Gaussian process on
the circle
of perimeter one. We prove a universality result for the shape of the
probability density function of the ground state energy
http://front.math.ucdavis.edu/math.PR/0611555
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4881. ON THE ESTIMATES OF THE DENSITY OF FEYNMAN-KAC SEMIGROUPS OF $
\ALPHA$-STABLE-LIKE PROCESSES
Chunlin Wang
Suppose that $\alpha \in (0,2)$ and that $X$ is an $\alpha$-stable-like
process on $\R^d$. Let $F$ be a function on $\R^d$ belonging to the
class
$\bf{J_{d,\alpha}}$ (see Introduction) and $A_{t}^{F}$ be $\sum_{s \le
t}F(X_{s-},X_{s}), t> 0$, a discontinuous additive functional of $X$.
With
neither $F$ nor $X$ being symmetric, under certain conditions, we
show that the
Feynman-Kac semigroup $\{S_{t}^{F}:t \ge 0\}$ defined by $$
S_{t}^{F}f(x)=\mathbb{E}_{x}(e^{-A_{t}^{F}}f(X_{t}))$$ has a density
$q$ and
that there exist positive constants $C_1,C_2,C_3$ and $C_4$ such that
$$C_{1}e^{-C_{2}t}t^{-\frac{d}{\alpha}}(1 \wedge
\frac{t^{\frac{1}{\alpha}}}{|x-y|})^{d+\alpha} \leq q(t,x,y) \leq
C_{3}e^{C_{4}t}t^{-\frac{d}{\alpha}}(1 \wedge
\frac{t^{\frac{1}{\alpha}}}{|x-y|})^{d+\alpha}$$ for all $(t,x,y)\in
(0,\infty)
\times \R^d \times \R^d$.
http://front.math.ucdavis.edu/math.PR/0611565
---------------------------------------------------------------
4882. ON THE ESTIMATES OF THE DENSITY OF THE PURELY DISCONTINUOUS
GIRSANOV TRANSFORMS OF $\ALPHA$-STABLE-LIKE PROCESSES
Chunlin Wang
In this paper, we study the purely discontinuous Girsanov transforms
which
were discussed in Chen and Song \cite{CS2} and Song \cite{S3}. We
show that the
transition density of any purely discontinuous Girsanov transform of a
$\alpha$-stable-like process, which can be nonsymmetric, is
comparable to the
transition density of the $\alpha$-stable-like process.
http://front.math.ucdavis.edu/math.PR/0611566
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4883. ON THE SPEED OF A COOKIE RANDOM WALK
Anne-Laure Basdevant (PMA) and Arvind Singh (PMA)
We consider the model of the one-dimensional cookie random walk when the
initial cookie distribution is spatially uniform and the number of
cookies per
site is finite. We give a criterion to decide whether the limiting
speed of the
walk is non-zero. In particular, we show that a positive speed may be
obtained
for just 3 cookies per site. We also prove a result on the continuity
of the
speed with respect to the initial cookie distribution.
http://front.math.ucdavis.edu/math.PR/0611580
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4884. THE NORM OF PRODUCTS OF FREE RANDOM VARIABLES
Vladislav Kargin
Let $X_i$ denote free identically-distributed random variables. This
paper
investigates how the norm of products $\Pi_n=X_1 X_2 ... X_n$ behaves
as $n$
approaches infinity. In addition, for positive $X_i$ it studies the
asymptotic
behavior of the norm of $Y_n=X_1 \circ X_2 \circ ...\circ X_n$, where
$\circ$
denotes the symmetric product of two positive operators: $A \circ
B=:A^{1/2}BA^{1/2}$.
It is proved that if the expectation of $X_i$ is 1, then the norm
of the
symmetric product $Y_{n}$ is between $c_1 n^{1/2}$ and $c_2 n$ for
certain
constant $c_1$ and $c_2$. That is, the growth in the norm is at most
linear.
For the norm of the usual product $Pi_n$, it is proved that the
limit of
$n^{-1}\log Norm(Pi_n)$ exists and equals $\log \sqrt{E(X_i^{\ast}X_
{i})}.$ In
other words, the growth in the norm of the product is exponential and
the rate
equals the logarithm of the Hilbert-Schmidt norm of operator X.
Finally, if $\pi $ is a cyclic representation of the algebra
generated by
$X_i$, and if $\xi$ is a cyclic vector, then $n^{-1}\log Norm(\pi
(\Pi_{n})
\xi)=\log \sqrt{E(X_{i}^{\ast}X_{i})}$ for all $n.$ In other words,
the growth
in the length of the cyclic vector is exponential and the rate
coincides with
the rate in the growth of the norm of the product.
These results are significantly different from analogous results for
commuting random variables and generalize results for random matrices
derived
by Kesten and Furstenberg.
http://front.math.ucdavis.edu/math.PR/0611593
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4885. HOMOGENIZATION OF SPACE-TIME DEPENDENT AND DEGENERATE RANDOM FLOWS
R\'{e}mi Rhodes (LATP)
We study a diffusion process with random space-time dependent
coefficients.
Moreover the diffusion matrix is allowed to degenerate. An invariance
principle
is proved provided that the diffusion coefficient is controlled by a
time
independent one.
http://front.math.ucdavis.edu/math.PR/0611598
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4886. ON LAWS OF LARGE NUMBERS FOR RANDOM WALKS
Anders Karlsson and Fran\c{c}ois Ledrappier
We prove a general noncommutative law of large numbers. This applies in
particular to random walks on any locally finite homogeneous graph,
as well as
to Brownian motion on Riemannian manifolds which admit a compact
quotient. It
also generalizes Oseledec's multiplicative ergodic theorem. In
addition, we
show that $\epsilon$-shadows of any ballistic random walk with finite
moment on
any group eventually intersect. Some related results concerning
Coxeter groups
and mapping class groups are recorded in the last section.
http://front.math.ucdavis.edu/math.PR/0611607
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4887. RANDOM SAMPLING IN CHIRP SPACE
Eric Carlen and R. Vilela Mendes
For the space of functions that can be approximated by linear chirps, we
prove a reconstruction theorem by random sampling at arbitrary rates.
http://front.math.ucdavis.edu/math.PR/0611608
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4888. QUENCHED INVARIANCE PRINCIPLES FOR RANDOM WALKS WITH RANDOM
CONDUCTANCES
P. Mathieu
We prove an almost sure invariance principle for a random walker
among i.i.d.
conductances in $\Z^d$, $d\geq 2$. We assume conductances are bounded
from
above but we dot require they are bounded from below.
http://front.math.ucdavis.edu/math.PR/0611613
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4889. ESTIMATION OF PARAMETERS OF STABLE DISTRIBUTIONS
Chunlin Wang
In this paper, we propose a method based on GMM (the generalized
method of
moments) to estimate the parameters of stable distributions with $0<
\alpha<2$.
We don't assume symmetry for stable distributions.
http://front.math.ucdavis.edu/math.ST/0611567
---------------------------------------------------------------
4890. HIGH DIMENSIONAL STATISTICAL INFERENCE AND RANDOM MATRICES
Iain M. Johnstone
Multivariate statistical analysis is concerned with observations on
several
variables which are thought to possess some degree of inter-
dependence. Driven
by problems in genetics and the social sciences, it first flowered in
the
earlier half of the last century. Subsequently, random matrix theory
(RMT)
developed, initially within physics, and more recently widely in
mathematics.
While some of the central objects of study in RMT are identical to
those of
multivariate statistics, statistical theory was slow to exploit the
connection.
However, with vast data collection ever more common, data sets now
often have
as many or more variables than the number of individuals observed. In
such
contexts, the techniques and results of RMT have much to offer
multivariate
statistics. The paper reviews some of the progress to date.
http://front.math.ucdavis.edu/math.ST/0611589
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4891. RANDOM WALKS ON DIRECTED CAYLEY GRAPHS
Ravi Montenegro
Previous authors have shown bounds on mixing time of random walks on
finite
undirected Cayley graphs, both with and without self-loops. We extend
this to
the most general case by showing that a similar bound holds for walks
on all
finite directed Cayley graphs. These are shown by use of two new
canonical path
theorems for mixing time of non-reversible Markov chains. The first
result is
related to the traditional canonical path mixing result but holds for
general
walks with small holding probability. The second theorem holds for
all finite
Markov chains, even non-reversible walks with no holding probability.
Curiously, these results are shown by use of Evolving sets, whereas
previous
path results were shown via Spectral gap.
http://front.math.ucdavis.edu/math.CO/0611585
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4892. POWER-LAW ESTIMATES FOR THE CENTRAL LIMIT THEOREM FOR CONVEX SETS
B. Klartag
We investigate the rate of convergence in the central limit theorem for
convex sets. We obtain bounds with a power-law dependence on the
dimension.
These bounds are asymptotically better than the logarithmic estimates
which
follow from the original proof of the central limit theorem for
convex sets.
http://front.math.ucdavis.edu/math.MG/0611577
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4893. A CANONICAL PATH APPROACH TO BOUNDING COLLISION TIME FOR
POLLARD'S RHO ALGORITHM
Ravi Montenegro
We show how to apply the canonical path method to a non-reversible
Markov
chain with no holding probability: a random walk used in Pollard's Rho
algorithm for discrete logarithm. This is used to show that the
Pollard Rho
method for finding the discrete logarithm on a cyclic group $G$ requires
$O(\sqrt{|G|} (\log |G|)^{3/2})$ steps until a collision occurs and
discrete
logarithm is possibly found, not far from the widely conjectured
value of
$\Theta(\sqrt{|G|})$. Conversely, we find that arguments based on
spectral gap,
spectral profile or log-Sobolev cannot be used to show the correct
mixing bound
of the Pollard Rho walk, while coupling can give at best a small
improvement on
our current bound for collision time.
http://front.math.ucdavis.edu/math.NT/0611586
---------------------------------------------------------------
4894. ON THE EQUIVALENCE OF SOME ETERNAL ADDITIVE COALESCENTS
Anne-Laure Basdevant (PMA)
In this paper, we study additive coalescents. Using their
representation as
fragmentation processes, we prove that the law of a large class of
eternal
additive coalescents is absolutely continuous with respect to the law
of the
standard additive coalescent on any bounded time interval.
http://front.math.ucdavis.edu/math.PR/0611523
---------------------------------------------------------------
4895. JOINT DENSITY FOR THE LOCAL TIMES OF CONTINUOUS-TIME MARKOV CHAINS
David Brydges and Remco van der Hofstad and Wolfgang K\"onig
We investigate the local times of a continuous-time Markov chain on an
arbitrary discrete state space. For fixed finite range of the Markov
chain, we
derive an explicit formula for the joint density of all local times
on the
range, at any fixed time. We use standard tools from the theory of
stochastic
processes and finite-dimensional complex calculus.
We apply this formula in the following directions: (1) we derive
large
deviation upper estimates for the normalized local times beyond the
exponential
scale, (2) we derive the upper bound in Varadhan's lemma for any
measurable
functional of the local times, \ch{and} (3) we derive large deviation
upper
bounds for continuous-time simple random walk on large subboxes of $
\Z^d$
tending to $\Z^d$ as time diverges. We finally discuss the relation
of our
density formula to the Ray-Knight theorem for continuous-time simple
random
walk on $\Z$, which is analogous to the well-known Ray-Knight
description of
Brownian local times.
http://front.math.ucdavis.edu/math.PR/0611525
---------------------------------------------------------------
4896. A NOTE ON INSENSITIVITY IN STOCHASTIC NETWORKS
Stan Zachary
We give a simple and direct treatment of insensitivity in stochastic
networks
which is quite general and which provides probabilistic insight into the
phenomenon. In the case of multi-class networks, the results
generalise those
of Bonald and Proutiere (2002, 2003).
http://front.math.ucdavis.edu/math.PR/0611526
---------------------------------------------------------------
4897. MARCHE AL\'{E}ATOIRE SUR UN IMMEUBLE AFFINE DE TYPE $\TILDE{A}_R
$ ET MOUVEMENT BROWNIEN DE LA CHAMBRE DE WEYL
Bruno Schapira (MAPMO and PMA)
In this paper we study a random walk on an affine building, whose radial
part, when suitably normalized, converges to the Brownian motion of
the Weyl
chamber (for the type $A$). This gives a new discrete approximation
of this
process, which is different from the one of Biane \cite{Bia2}. The main
ingredients of the proof are a combinatorial formula on the building
and the
estimate of the transition density proved in \cite{AST}. Moreover our
result
extends in higher rank the correspondence at the probabilistic level
between
Riemannian symmetric spaces of the noncompact type and their discrete
counterpart, which had been previously obtained by Bougerol and
Jeulin in rank
one \cite{BJ}.
http://front.math.ucdavis.edu/math.PR/0611529
---------------------------------------------------------------
4898. MULTISCALE ANALYSIS FOR SPDES WITH QUADRATIC NONLINEARITIES
D. Bloemker and G.A. Pavliotis and M. Hairer
In this article we derive rigorously amplitude equations for
stochastic PDEs
with quadratic nonlinearities, under the assumption that the noise
acts only on
the stable modes and for an appropriate scaling between the distance
from
bifurcation and the strength of the noise. We show that, due to the
presence of
two distinct timescales in our system, the noise (which acts only on
the fast
modes) gets transmitted to the slow modes and, as a result, the
amplitude
equation contains both additive and multiplicative noise.
As an application we study the case of the one dimensional Burgers
equation
forced by additive noise in the orthogonal subspace to its dominant
modes. The
theory developed in the present article thus allows to explain
theoretically
some recent numerical observations from [Rob03].
http://front.math.ucdavis.edu/math.PR/0611537
---------------------------------------------------------------
4899. COHERENT RANDOM PERMUTATIONS WITH RECORD STATISTICS
Alexander Gnedin
Random permutations with distribution conditionally uniform given the
set of
record values can be generated in a unified way, coherently for all
values of
$n$. Our central example is a two-parameter family of random
permutations that
are conditionally uniform given the counts of upper and lower
records. This
family interpolates between two versions of Ewens' distribution. We
discuss
characterisations of the conditionally uniform permutations, their
asymptotic
properties, constructions and relations to random partitions.
http://front.math.ucdavis.edu/math.PR/0611538
---------------------------------------------------------------
4900. CYCLES OF FREE WORDS IN SEVERAL INDEPENDENT RANDOM PERMUTATIONS
Florent Benaych-Georges (PMA)
In this text, extending results of A.Nica and M. Neagu, we study the
asymptotics of the number of cycles of a given length of a word in
several
independent random permutations with restricted cycle lengths.
Specifically,
for $A_1$,..., $A_k$ non empty sets of positive integers and for $w$
word in
the letters $g_1,g_1^{-1}$,..., $g_k,g_k^{-1}$, we consider, for all
$n$ such
that it is possible, an independent family $s_1(n)$,..., $s_k(n)$ of
random
permutations chosen uniformly among the permutations of $n$ objects
which have
all their cycle lengths in respectively $A_1$,..., $A_k$, and for $l$
positive
integer, we are going to give asymptotics (as $n$ goes to infinity)
on the
number $N_l(n)$ of cycles of length $l$ of the permutation obtained
by changing
any letter $g_i$ in $w$ by $s_i(n)$. In many cases, we prove that the
distribution of $N_l(n)$ converges to a Poisson law with parameter $1/
l$ and
that the family of random variables $(N_1(n), N_2(n),...)$ is
asymptotically
independent. We notice the pretty surprising fact that from this
point of view,
many things happen like if we considered the number of cycles of
given lengths
of a single permutation with uniform distribution on the $n$-th
symmetric
group.
http://front.math.ucdavis.edu/math.PR/0611500
---------------------------------------------------------------
4901. CENTRAL LIMIT THEOREM FOR A TAGGED PARTICLE IN ASYMMETRIC
SIMPLE EXCLUSION
Patricia Goncalves
We prove a Functional Central Limit Theorem for the position of a Tagged
Particle in the one-dimensional Asymmetric Simple Exclusion Process
in the
hyperbolic scaling, starting from a Bernoulli product measure
conditioned to
have a particle at the origin. We also prove that the position of the
Tagged
Particle at time $t$ depends on the initial configuration, by the
number of
empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$
in the
hyperbolic and in a longer time scale, namely $N^{4/3}$.
http://front.math.ucdavis.edu/math.PR/0611505
---------------------------------------------------------------
4902. QUANTUM STOCHASTIC CONVOLUTION COCYCLES II
J.Martin Lindsay and Adam Skalski
The theory of quantum stochastic convolution cocycles is extended to the
topological context of compact quantum groups. Stochastic convolution
cocycles
on a C*-hyperbialgebra, which are Markov-regular, completely positive
and
contractive, are shown to satisfy coalgebraic quantum stochastic
differential
equations with completely bounded coefficients, and the structure of
their
stochastic generators is obtained. Specialising to *-homomorphic
convolution
cocycles on a C*-bialgebra the stochastic generators are shown to have
Schuermann form. Tentative definitions of quantum Levy process on a
compact
quantum group, for which a reconstruction theorem is valid, are
proposed. In
the examples given, connection to the algebraic theory is emphasised
by a focus
on the case of full compact quantum groups.
http://front.math.ucdavis.edu/math.OA/0611497
---------------------------------------------------------------
4903. HOW OFTEN IS THE COORDINATE OF A HARMONIC OSCILLATOR POSITIVE?
Boris Tsirelson
The coordinate of a harmonic oscillator is measured at a time chosen at
random among three equiprobable instants: now, after one third of the
period,
or after two thirds. The (total) probability that the outcome is
positive
depends on the state of the oscillator. In the classical case the
probability
varies between 1/3 and 2/3, but in the quantum case -- between 0.29
and 0.71.
http://front.math.ucdavis.edu/quant-ph/0611147
---------------------------------------------------------------
4904. RANDOM GRAPH-HOMOMORPHISMS AND LOGARITHMIC DEGREE
Itai Benjamini and Ariel Yadin and Amir Yehudayoff
A graph homomorphism between two graphs is a map from the vertex set
of one
graph to the vertex set of the other graph, that maps edges to edges.
In this
note we study the range of a uniformly chosen homomorphism from a
graph G to
the infinite line Z. It is shown that if the maximal degree of G is
`sub-logarithmic', then the range of such a homomorphism is super-
constant.
Furthermore, some examples are provided, suggesting that perhaps
for graphs
with super-logarithmic degree, the range of a typical homomorphism is
bounded.
In particular, a sharp transition is shown for a specific family of
graphs
C_{n,k} (which is the tensor product of the n-cycle and a complete
graph, with
self-loops, of size k). That is, given any function psi(n) tending to
infinity,
the range of a typical homomorphism of C_{n,k} is super-constant for
k = 2
log(n) - psi(n), and is 3 for k = 2 log(n) + psi(n).
http://front.math.ucdavis.edu/math.PR/0611416
---------------------------------------------------------------
4905. ON PENROSE'S SQUARE-ROOT LAW AND BEYOND
Werner Kirsch
In certain bodies, like the Council of the EU, the member states have a
voting weight which depends on the population of the re- spective
state. In
this article we ask the question which voting weight guarantees a `fair'
representation of the citizens in the union. The tra- ditional
answer, the
square-root law by Penrose, is that the weight of a state (more
precisely: the
voting power) should be proportional to the square-root of the
population of
this state. The square root law is based on the assumption that the
voters in
every state cast their vote inde- pendently of each other. In this
paper we
concentrate on cases where the independence assumption is not valid.
http://front.math.ucdavis.edu/math.PR/0611418
---------------------------------------------------------------
4906. ON A MODEL FOR THE STORAGE OF FILES ON A HARDWARE I :
STATISTICS AT A FIXED TIME AND ASYMPTOTICS
Vincent Bansaye (PMA)
We consider a generalized version in continuous time of the parking
problem
of Knuth. Files arrive following a Poisson point process and are
stored on a
hardware identified with the real line. We specify the distribution
of the
space of unoccupied locations at a fixed time and give its
asymptotics when the
hardware is becoming full.
http://front.math.ucdavis.edu/math.PR/0611432
---------------------------------------------------------------
4907. EXISTENCE OF SUBCRITICAL REGIMES IN THE POISSON BOOLEAN MODEL
OF CONTINUUM PERCOLATION
Jean-Baptiste Gou\'er\'e (MAPMO)
We consider the so-called Poisson Boolean model of continuum
percolation. At
each point of an homogeneous Poisson point process on the Euclidean
space
$\R^d$, we center a ball with random radius. We assume that the radii
of the
balls are independent, identically distributed and independent of the
point
process. We denote by $\Sigma$ the union of the balls and by $S$ the
connected
component of $\Sigma$ that contains the origin. We show that $S$ is
almost
surely bounded for small enough density $\lambda$ of the point
process if and
only if the mean volume of the balls is finite. Let us denote by $D$ the
diameter of $S$ and by $R$ one of the random radii. We also show
that, for all
positive real number $s$, $D^s$ is integrable for small enough $
\lambda$ if and
only if $R^{d+s}$ is integrable.
http://front.math.ucdavis.edu/math.PR/0611369
---------------------------------------------------------------
4908. ON THE EXISTENCE OF SOME ARCH($\INFTY$) PROCESSES
Philippe Soulier (MODAL'X) and Randal Douc (CMAP) and Fran\c{c}ois
Roueff (LTCI)
A new sufficient condition for the existence of a stationary causal
solution
of an ARCH($\infty$) equation is provided. This condition allows to
consider
polynomially decaying coefficients, so that it can be applied to the
so-called
FIGARCH processes, whose existence is thus proved.
http://front.math.ucdavis.edu/math.ST/0611339
---------------------------------------------------------------
4909. TREES AND ASYMPTOTIC DEVELOPMENTS FOR FRACTIONAL STOCHASTIC
DIFFERENTIAL EQUATIONS
Andreas Neuenkirch and Ivan Nourdin (PMA) and Andreas Roessler and
Samy Tindel (IECN)
In this paper we consider a n-dimensional stochastic differential
equation
driven by a fractional Brownian motion with Hurst parameter H>1/3. After
solving this equation in a rather elementary way, following the
approach of
Gubinelli, we show how to obtain an expansion for E[f(X\_t)] in terms
of t,
where X denotes the solution to the SDE and f:R^n->R is a regular
function.
With respect to the work by Baudoin and Coutin, where the same kind
of problem
is considered, we try an improvement in three different directions:
we are able
to take a drift into account in the equation, we parametrize our
expansion with
trees (which makes it easier to use), and we obtain a sharp control
of the
remainder.
http://front.math.ucdavis.edu/math.PR/0611306
---------------------------------------------------------------
4910. A NEAR NEIGHBOUR CONTINUUM PERCOLATION MODEL
A. Gillett and M. Nuyens
We introduce a continuum percolation model defined on the points of a
d-dimensional homogeneous Poisson process. Each Poisson point is
connected to
all points within its connection range, which depends on the
distances to the
other Poisson points. We show that the new model exhibits a phase
transition,
and obtain results about the critical values in low and high dimensions.
http://front.math.ucdavis.edu/math.PR/0611315
---------------------------------------------------------------
4911. RANDOM DISCRETE MATRICES
V. Vu
In this survey, we discuss some basic problems concerning random
matrices
with discrete distributions. Several new results, tools and
conjectures will be
presented.
http://front.math.ucdavis.edu/math.CO/0611321
---------------------------------------------------------------
4912. APPROXIMATING GENEALOGIES FOR PARTIALLY LINKED NEUTRAL LOCI
UNDER A SELECTIVE SWEEP
P. Pfaffelhuber and A. Studeny
Consider a genetic locus carrying a strongly beneficial allele which has
recently fixed in a large population. As strongly beneficial alleles fix
quickly, sequence diversity at partially linked neutral loci is
reduced. This
phenomenon is known as a selective sweep. The fixation of the
beneficial allele
not only affects sequence diversity at single neutral loci but also
the joint
allele distribution of several partially linked neutral loci. This
distribution
can be studied using the ancestral recombination graph for samples of
partially
linked neutral loci during the selective sweep. To approximate this
graph, we
extend recent work by Schweinsberg & Durrett 2005 and Etheridge,
Pfaffelhuber &
Wakolbinger 2006 using a marked Yule tree for the genealogy at a
single neutral
locus linked to a strongly beneficial one. We focus on joint
genealogies at two
partially linked neutral loci in the case of large selection
coefficients
\alpha and recombination rates \rho = O(\alpha/\log\alpha) between
loci. Our
approach leads to a full description of the genealogy with accuracy
of O((\log
\alpha)^{-2}) in probability. As an application, we derive the
expectation of
Lewontin's D as a measure for non-random association of alleles.
http://front.math.ucdavis.edu/q-bio.PE/0611029
---------------------------------------------------------------
4913. QUASI-ARITHMETIC MEANS OF COVARIANCE FUNCTIONS WITH POTENTIAL
APPLICATIONS TO SPACE-TIME DATA
E. Porcu and J. Mateu and and G. Christakos
The theory of quasi-arithmetic means is a powerful tool in the study of
covariance functions across space-time. In the present study we use
quasi-arithmetic functionals to make inferences about the
permissibility of
averages of functions that are not, in general, permissible covariance
functions. This is the case, e.g., of the geometric and harmonic
averages, for
which we obtain permissibility criteria. Also, some important
inequalities
involving covariance functions and preference relations as well as
algebraic
properties can be derived by means of the proposed approach. In
particular, we
show that quasi-arithmetic covariances allow for ordering and preference
relations, for a Jensen-type inequality and for a minimal and maximal
element
of their class. The general results shown in this paper are then
applied to
study of spatial and spatiotemporal random fields. In particular, we
discuss
the representation and smoothness properties of a weakly stationary
random
field with a quasi-arithmetic covariance function. Also, we show that
the
generator of the quasi-arithmetic means can be used as a link
function in order
to build a space-time nonseparable structure starting from the
spatial and
temporal margins, a procedure that is technically sound for those
working with
copulas. Several examples of new families of stationary covariances
obtainable
with this procedure are shown. Finally, we use quasi-arithmetic
functionals to
generalise existing results concerning the construction of nonstationary
spatial covariances and discuss the applicability and limits of this
generalisation.
http://front.math.ucdavis.edu/math.PR/0611275
---------------------------------------------------------------
4914. CONSISTENT FAMILIES OF BROWNIAN MOTIONS AND STOCHASTIC FLOWS OF
KERNELS
Chris Howitt and Jon Warren
Consider the following mechanism for the random evolution of a
distribution
of mass on the integer lattice. At unit rate, independently for each
site, the
mass at the site is split into two parts by choosing a random proportion
distributed according to some specified probability measure on [0,1] and
dividing the mass in that proportion. One part then moves to each of
the two
adjacent sites. This paper considers a continuous analogue of this
evolution,
which may be described by means of a stochastic flow of kernels, the
theory of
which was developed by Le Jan and Raimond. One of their results is
that such a
flow is characterized by specifying its N point motions, which form a
consistent family of Brownian motions. This means for each N we have a
diffusion in N dimensional Euclidean space, whose N co-ordinates are all
Brownian motions. Any M co-ordinates taken from the N-dimensional
process are
distributed as the M-dimensional process in the family. Moreover, in
this
setting, the only interactions between co-ordinates are local: when
coordinates
differ in value they evolve independently of each other. In this
paper we
explain how such multidimensional diffusions may be constructed and
characterized via martingale problems.
http://front.math.ucdavis.edu/math.PR/0611292
---------------------------------------------------------------
4915. SIMPLE MONTE CARLO AND THE METROPOLIS ALGORITHM
Peter Mathe and Erich Novak
We study the integration of functions with respect to an unknown
density. We
compare the simple Monte Carlo method (which is almost optimal for a
certain
large class of inputs) and compare it with the Metropolis algorithm
(based on a
suitable ball walk).
Using MCMC we prove (for certain classes of inputs) that adaptive
methods are
much better than nonadaptive ones. Actually, the curse of dimension (for
nonadaptive methods) can be broken by adaption.
http://front.math.ucdavis.edu/math.NA/0611285
---------------------------------------------------------------
4916. COMPLETELY POSITIVE QUANTUM STOCHASTIC CONVOLUTION COCYCLES AND
THEIR DILATIONS
Adam Skalski
Stochastic generators of completely positive and contractive quantum
stochastic convolution cocycles on a C*-hyperbialgebra are
characterised. The
characterisation is used to obtain dilations and stochastic forms of
Stinespring decomposition for completely positive convolution
cocycles on a
C*-bialgebra.
http://front.math.ucdavis.edu/math.OA/0611271
---------------------------------------------------------------
4917. A NEW METHOD OF NORMAL APPROXIMATION. I. GEOMETRIC CENTRAL
LIMIT THEOREMS
Sourav Chatterjee
We introduce a new version of Stein's method that reduces a large
class of
normal approximation problems to variance bounding exercises, thus
making a
connection between central limit theorems and concentration of
measure. Unlike
Skorokhod embeddings, the object whose variance has to be bounded has an
explicit formula that makes it possible to carry out the program more
easily.
As an application, we derive a general CLT for functions that are
obtained as
combinations of many local contributions, where the definition of
`local'
itself depends on the data. Several examples are given, including the
solution
to a nearest-neighbor CLT problem posed by Peter Bickel.
http://front.math.ucdavis.edu/math.PR/0611213
---------------------------------------------------------------
4918. HITTING TIME OF LARGE SUBSETS OF THE HYPERCUBE
Jiri Cerny and Veronique Gayrard
We study the simple random walk on the $n$-dimensional hypercube, in
particular its hitting times of large (possibly random) sets. We give
simple
conditions on these sets ensuring that the properly-rescaled hitting
time is
asymptotically exponentially distributed, uniformly in the starting
position of
the walk. These conditions are then verified for percolation clouds with
densities that are much smaller than $(n \log n)^{-1}$. A main
motivation
behind this paper is the study of the so-called aging phenomenon in
the Random
Energy Model (REM), the simplest model of a mean-field spin glass.
Our results
allow us to prove aging in the REM for all temperatures, thereby
extending
earlier results to their optimal temperature domain.
http://front.math.ucdavis.edu/math.PR/0611242
---------------------------------------------------------------
4919. A CHARACTERIZATION OF HARMONIC MEASURES ON LAMINATIONS BY
HYPERBOLIC RIEMANN SURFACES
Yuri Bakhtin and Matilde Martinez
We prove that a probability measure on a compact non-singular
lamination by
hyperbolic Riemann surfaces is harmonic if and only if it is the
projection of
a measure on the unit tangent bundle such that it is invariant under
both the
geodesic and the horocycle flows.
http://front.math.ucdavis.edu/math.DS/0611235
---------------------------------------------------------------
4920. CONVERGENCE OF THE LENGTH OF THE LOOP-ERASED RANDOM WALK ON
FINITE GRAPHS TO THE RAYLEIGH PROCESS
Jason Schweinsberg
Let $(G_n)_{n=1}^{\infty}$ be a sequence of finite graphs, and let
$Y_t$ be
the length of a loop-erased random walk on $G_n$ after $t$ steps. We
show that
for a large family of sequences of finite graphs, which includes the
case in
which $G_n$ is the $d$-dimensional torus of size-length $n$ for $d
\geq 4$, the
process $(Y_t)_{t=0}^{\infty}$, suitably normalized, converges to the
Rayleigh
process introduced by Evans, Pitman, and Winter. Our proof relies
heavily on
ideas of Peres and Revelle, who used loop-erased random walks to show
that the
uniform spanning tree on large finite graphs converges to the Brownian
continuum random tree of Aldous.
http://front.math.ucdavis.edu/math.PR/0611155
---------------------------------------------------------------
4921. HEIGHT PROCESS FOR SUPER-CRITICAL CONTINUOUS STATE BRANCHING
PROCESS
Jean-Fran\c{c}ois Delmas (CERMICS)
We define the height process for super-critical continuous state
branching
processes with quadratic branching mechanism. It appears as a
projective limit
of Brownian motions with positive drift reflected at 0 and a>0 as a
goes to
infinity. Then we extend the pruning procedure of branching processes
to the
super-critical case. This give a complete duality picture between
pruning and
size proportional immigration for quadratic continuous state branching
processes.
http://front.math.ucdavis.edu/math.PR/0611172
---------------------------------------------------------------
4922. ELEMENTARY POTENTIAL THEORY ON THE HYPERCUBE
Gerard Ben Arous and Veronique Gayrard
This work addresses potential theoretic questions for the standard
nearest
neighbor random walk on the hypercube $\{-1,+1\}^N$. For a large
class of
subsets $A\subset\{-1,+1\}^N$ we give precise estimates for the harmonic
measure of $A$, the mean hitting time of $A$, and the Laplace
transform of this
hitting time. In particular, we give precise sufficient conditions
for the
harmonic measure to be asymptotically uniform, and for the hitting
time to be
asymptotically exponentially distributed, as $N\to\infty$. Our
approach relies
on a $d$-dimensional extension of the Ehrenfest urn scheme called
lumping and
covers the case where $d$ is allowed to diverge with $N$ as long as
$d\leq\alpha_0\frac{N}{\log N}$ for some constant $0<\alpha_0<1$.
http://front.math.ucdavis.edu/math.PR/0611178
---------------------------------------------------------------
4923. DIRECTED ANIMALS IN THE GAS
Yvan Le Borgne (LaBRI) and Jean-Fran\c{c}ois Marckert (LaBRI)
In this paper, we revisit the enumeration of directed animals using gas
models. We show that there exists a natural construction of random
directed
animals on any directed graph together with a particle system that
explains at
the level of objects the formal link known between the density of the
gas model
and the generating function of directed animals counted according to
the area.
This provides some new methods to compute the generating function of
directed
animals counted according to area, and leads in the particular case
of the
square lattice to new combinatorial results and questions. A model of
gas
related to directed animals counted according to area and perimeter
on any
directed graph is also exhibited.
http://front.math.ucdavis.edu/math.PR/0611194
---------------------------------------------------------------
4924. KDV PRESERVES WHITE NOISE
Jeremy Quastel and Benedek Valko
It is shown that white noise is an invariant measure for the Korteweg-
deVries
equation on $\mathbb T$. This is a consequence of recent results of
Kappeler
and Topalov establishing the well-posedness of the equation on
appropriate
negative Sobolev spaces, together with a result of
Cambronero and McKean that white noise is the image under the
Miura transform
(Ricatti map) of the (weighted) Gibbs measure for the modified KdV
equation,
proven to be invariant for that equation by Bourgain.
http://front.math.ucdavis.edu/math.AP/0611152
---------------------------------------------------------------
4925. MUTATION, SELECTION, AND ANCESTRY IN BRANCHING MODELS: A
VARIATIONAL APPROACH
Ellen Baake and Hans-Otto Georgii
We consider the evolution of populations under the joint action of
mutation
and differential reproduction, or selection. The population is
modelled as a
finite-type Markov branching process in continuous time, and the
associated
genealogical tree is viewed both in the forward and the backward
direction of
time. The stationary type distribution of the reversed process, the
so-called
ancestral distribution, turns out as a key for the study of mutation-
selection
balance. This balance can be expressed in the form of a variational
principle
that quantifies the respective roles of reproduction and mutation for
any
possible type distribution. It shows that the mean growth rate of the
population results from a competition for a maximal long-term growth
rate, as
given by the difference between the current mean reproduction rate,
and an
asymptotic decay rate related to the mutation process; this tradeoff
is won by
the ancestral distribution.
Our main application is the quasispecies model of sequence
evolution with
mutation coupled to reproduction but independent across sites, and a
fitness
function that is invariant under permutation of sites. Here, the
variational
principle is worked out in detail and yields a simple, explicit result.
http://front.math.ucdavis.edu/q-bio.PE/0611018
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4926. DISCRETE APPROXIMATIONS TO REFLECTED BROWNIAN MOTION
Krzysztof Burdzy and Zhen-Qing Chen
In this paper we investigate three discrete or semi-discrete
approximation
schemes for reflected Brownian motion on bounded Euclidean domains.
http://front.math.ucdavis.edu/math.PR/0611114
---------------------------------------------------------------
4927. SLE(6) AND CLE(6) FROM CRITICAL PERCOLATION
Federico Camia and Charles M. Newman
We review some of the recent progress on the scaling limit of two-
dimensional
critical percolation; in particular, the convergence of the
exploration path to
chordal SLE(6) and the "full" scaling limit of cluster interface
loops. The
results given here on the full scaling limit and its conformal
invariance
extend those presented previously. For site percolation on the
triangular
lattice, the results are fully rigorous. We explain some of the main
ideas,
skipping most technical details.
http://front.math.ucdavis.edu/math.PR/0611116
---------------------------------------------------------------
4928. SOME ESTIMATES FOR PLANAR RANDOM WALK AND BROWNIAN MOTION
Christian Benes
The purpose of this note is to collect in one place a few results about
simple random walk and Brownian motion which are often useful. These
include
standard results such as Beurling estimates, large deviation
estimates, and a
method for coupling the two processes, as well as solutions to the
discrete
Dirichlet problem in various domains which, to the author's
knowledge, have not
been published anywhere. The main focus is on the two-dimensional
processes.
http://front.math.ucdavis.edu/math.PR/0611127
---------------------------------------------------------------
4929. NON-EXISTENCE OF RANDOM GRADIENT GIBBS MEASURES IN CONTINUOUS
INTERFACE MODELS IN $D=2$
A. C. D. van Enter and C. Kuelske
We consider statistical mechanics models of continuous spins in a
disordered
environment. These models have a natural interpretation as effective
interface
models. It is well-known that without disorder there are no interface
Gibbs
measures in infinite volume in dimension d=2, while there are
``gradient Gibbs
measures'' describing an infinite-volume distribution for the
increments of the
field, as was shown by Funaki and Spohn. In the present paper we show
that
adding a disorder term prohibits the existence of such gradient Gibbs
measures
for general interaction potentials in d=2. This non-existence result
generalizes the simple case of Gaussian fields where it follows from an
explicit computation. In d=3 where random gradient Gibbs measures are
expected
to exist, our method provides a lower bound of the order of the
inverse of the
distance on the decay of correlations of Gibbs expectations w.r.t. the
distribution of the random environment.
http://front.math.ucdavis.edu/math.PR/0611140
---------------------------------------------------------------
4930. PHASE TRANSITIONS FOR THE LONG-TIME BEHAVIOUR OF INTERACTING
DIFFUSIONS
A. Greven and F. den Hollander
This paper considers a system of interacting diffusions labelled by the
d-dimensional integer lattice. The diffusions interact linearly,
according to a
random walk transition kernel $a(\cdot,\cdot)$, and have an autonomous
quadratic diffusion function with diffusion parameter $b$. The ergodic
behaviour of such systems shows a delicate interplay between $a(cdot,
\cdot)$
and $b$, which is described in detail. For instance, when $a(\cdot,
\cdot)$ is
transient, there is a sequence of critical values $b_*>b_2>b_3>...$
at which
the equilibrium experiences a crossover: at $b_*$ from extinction to
survival,
and at $b_m$ from infinite to finite $m$-th moment. Proofs are based on
$L^2$-theory, large deviations, and Palm theory.
http://front.math.ucdavis.edu/math.PR/0611141
---------------------------------------------------------------
4931. COUNTING PLANAR RANDOM WALK HOLES
Christian Benes
We study two variants of the notion of {\it holes} formed by planar
simple
random walk of time duration $2n$ and the areas associated with them.
We prove
in both cases that the number of holes of area greater than A(n), where
$\{A(n)\}$ is an increasing sequence, is, up to a logarithmic
correction term,
asymptotic to $n\cdot A(n)^{-1}$ for a range of large holes, thus
confirming an
observation by Mandelbrot. A consequence is that the largest hole has
an area
which is logarithmically asymptotic to $n$. We also discuss the
different
exponent of 5/3 observed by Mandelbrot for small holes.
http://front.math.ucdavis.edu/math.PR/0611144
---------------------------------------------------------------
4932. PROBABILITY DISTRIBUTION OF DISTANCES BETWEEN LOCAL EXTREMA OF
RANDOM NUMBER SERIES
Argyn Kuketayev
There is a sequence of random numbers x1,x2, ..., xn and so on.
Numbers are
independent of each other, but all numbers are from the same continuous
distribution. If x1 < x2 > x3, then x2 is a local maximum. Here, we
show that
the probability mass function (PMF) of idstribution of distances
between local
maxima is non-parametric and the same for any probability
distribution of
random numbers in the sequence, and that the average distance is
exactly 3. We
present a method of computation of this PMF and its table for
distances betwen
2 and 29. This PMF is confirmed to match distance distributions of
sample
random number sequences, which were created by pseudo-random number
generators
or obtained from "true" random number sources.
http://front.math.ucdavis.edu/math.ST/0611130
---------------------------------------------------------------
4933. ON DISCRETIZATION SCHEMES FOR STOCHASTIC EVOLUTION EQUATIONS
Istvan Gy\"{o}ngy and Annie Millet (PMA and MATISSE and SAMOS)
Stochastic evolutional equations with monotone operators are
considered in
Banach spaces. Explicit and implicit numerical schemes are presented.
The
convergence of the approximations to the solution of the equations is
proved.
http://front.math.ucdavis.edu/math.PR/0611069
---------------------------------------------------------------
4934. COMPUTATION OF THE INVARIANT MEASURE FOR A L\'{E}VY DRIVEN SDE:
RATE OF CONVERGENCE
Fabien Panloup (PMA)
We study the rate of convergence of some recursive procedures based
on some
"exact" or "approximate" Euler schemes which converge to the
invariant measure
of an ergodic SDE driven by a L\'{e}vy process. The main interest of
this work
is to compare the rates induced by exact and approximate Euler
schemes. In our
main result, we show that replacing the small jumps by a Brownian
component in
the approximate case preserves the rate induced by the exact Euler
scheme for a
large class of L\'{e}vy processes.
http://front.math.ucdavis.edu/math.PR/0611072
---------------------------------------------------------------
4935. ON IMPLICIT AND EXPLICIT DISCRETIZATION SCHEMES FOR PARABOLIC
SPDES IN ANY DIMENSION
Annie Millet (MATISSE and Pma and Samos) and Pierre-Luc Morien
(MODAL'x)
We study the speed of convergence of the explicit and implicit space-
time
discretization schemes of the solution $u(t,x)$ to a parabolic partial
differential equation in any dimension perturbed by a space-
correlated Gaussian
noise. The coefficients only depend on $u(t,x)$ and the influence of the
correlation on the speed is observed.
http://front.math.ucdavis.edu/math.PR/0611073
---------------------------------------------------------------
4936. ON STRONGLY PETROVSKII'S PARABOLIC SPDES IN ARBITRARY DIMENSION
AND THE STOCHASTIC CAHN-HILLIARD EQUATION
Caoline Cardon-Weber (PMA) and Annie Millet (PMA and MATISSE and
SAMOS)
In this paper we show that the Cahn-Hilliard stochastic SPDE has a
function
valued solution in dimension 4 and 5 when the perturbation is driven
by a
space-correlated Gaussian noise. This is done proving general results
on SPDEs
with globally Lipschitz coefficients associated with operators on smooth
domains of $\mathbb{R}^d$ which are parabolic in the sense of
Petrovskii}, and
do not necessarily define a semi-group of operators. We study the
regularity of
the trajectories of the solutions and the absolute continuity of the
law at
some given time and position.
http://front.math.ucdavis.edu/math.PR/0611090
---------------------------------------------------------------
4937. WIGNER FORMULA OF ROTATION MATRICES AND QUANTUM WALKS
Takahiro Miyazaki and Makoto Katori and Norio Konno
Quantization of a random-walk model is performed by giving a multi-
component
qubit to a walker at site and by introducing a quantum coin, which is
represented by a unitary matrix. In quantum walks, the qubit of
walker is mixed
according to the quantum coin at each time step, when the walker hops
to other
sites. The standard (discrete) quantum-walk model in one-dimension is
defined
by using a $2 \times 2$ unitary matrix for a walker with two-
component qubit.
In this paper we use Wigner's $(2j+1)$-dimensional unitary
representations of
rotations as quantum coins, where $j$ is a half-integer, and
introduce a family
of one-dimensional quantum walks with $(2j+1)$-component qubits. For
any value
of half-integer $j$, convergence of all moments of walker's
pseudovelocity in
the long-time limit is proved. It is generally shown for the present
models
that, if $(2j+1)$ is even, the limit distribution is given by a
superposition
of $(2j+1)/2$ terms of scaled Konno's density functions, and if $(2j
+1)$ is
odd, it is a superposition of $j$ terms of scaled Konno's density
functions and
a Dirac's delta function at the origin. For the two-, three-, and
four-component models, the limit distribution functions are explicitly
calculated and their dependence on the parameters of quantum coins
and on the
initial qubit of walker is completely determined. Comparison with
computer
simulation results is also shown.
http://front.math.ucdavis.edu/quant-ph/0611022
---------------------------------------------------------------
4938. SIMULATION STUDIES OF SOME VORONOI POINT PROCESSES
K.A. Borovkov and D.A. Odell
We introduce a new class of dynamic point process models with simple and
intuitive dynamics that are based on the Voronoi tessellations
generated by the
processes. Under broad conditions, these processes prove to be
ergodic and
produce, on stabilisation, a wide range of clustering patterns. In
the paper,
we present results of simulation studies of three statistical
measures (Thiel's
redundancy, van Lieshout and Baddeley's $J$-function and the empirical
distribution of the Voronoi nearest neighbours' numbers) for
inference on these
models from the clustering behaviour in the stationary regime. In
particular,
we make comparisons with the area-interaction processes of Baddeley
and van
Lieshout.
http://front.math.ucdavis.edu/math.PR/0611031
---------------------------------------------------------------
4939. A NOTE ON THE EXCHANGEABILITY CONDITION IN STEIN'S METHOD
Adrian R\"ollin
We show by a surprisingly simple argument that the exchangeability
condition,
which is key to the exchangeable pair approach in Stein's method for
distributional approximation, can be omitted in many standard
settings. This is
achieved by replacing the usual antisymmetric function by a simpler
one, for
which only equality in distribution is required. In the case of
approximations
by continuous distributions we also slightly improve the constants
appearing in
previous results. For Poisson approximation, Chatterjee et al. (2005)
use a
different antisymmetric function, and additional error terms are
needed if
their bound is to be extended beyond the exchangeable setting. There
is a
strong connection between this new approach and Barbour's generator
interpretation of Stein's method.
http://front.math.ucdavis.edu/math.PR/0611050
---------------------------------------------------------------
4940. A NEW FORMULATION OF THE SPINE APPROACH TO BRANCHING DIFFUSIONS
Robert Hardy and Simon C. Harris
We present a formalization of the spine change of measure approach for
branching diffusions that improves on the scheme laid out for branching
Brownian motion in Kyprianou (2004) ["Travelling wave solutions to
the KPP
equation, Ann. Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72]
which
itself made use of earlier works of Lyons et al (1997) ["A conceptual
proof of
the Kesten-Stigum theorem for multi-type branching processes",
Classical and
modern branching processes, IMA Vol. Math. Appl., vol.84, Springer,
New York,
pp181-185]. We use our new formulation to interpret certain `Gibbs-
Boltzmann'
weightings of particles and use this to give a new, intuitive and
proof of a
more general `Many-to-One' result which enables expectations of sums
over
particles in the branching diffusion to be calculated purely in terms
of an
expectation of one particle. Significantly, our formalization has
provided the
foundations that facilitate a variety of new, greatly simplified and
more
intuitive proofs in branching diffusions: see, for example, the L^p
convergence
of additive martingales in Hardy and Harris (2006) ["Spine proofs for
L^p-convergence of branching-diffusion martingales", arXiv:math.PR/
0611056],
the path large deviation results for branching Brownian motion in
Hardy and
Harris (2006) ["A conceptual approach to a path result for branching
Brownian
motion", Stochastic Processes and their Applications,
doi:10.1016/j.spa.2006.05.010] and the large deviations for a
continuous-typed
branching diffusion in Git et al (2006) ["Exponential growth rates in
a typed
branching diffusion", Annals Applied Prob., (under revision)] and
Hardy and
Harris (2004) ["A spine proof of a lower-bound for a typed branching
diffusion", no.0408, Mathematics Preprint, University of Bath].
http://front.math.ucdavis.edu/math.PR/0611054
---------------------------------------------------------------
4941. SPINE PROOFS FOR LP-CONVERGENCE OF BRANCHING-DIFFUSION MARTINGALES
Robert Hardy and Simon C. Harris
Using the foundations laid down in Hardy and Harris (2006) ["A new
formulation of the spine approach in branching diffusions",
arXiv:math.PR/0611054], we present new spine proofs of the L^p-
convergence
p>=1) of some key `additive' martingales for three distinct models of
branching
diffusions, including new results for a multi-type branching Brownian
motion
and discussion of left-most particle speeds. The spine techniques we
develop
give clear and simple arguments in the spirit of the conceptual spine
proofs
found in Kyprianou (2004) ["Travelling wave solutions to the KPP
equation, Ann.
Inst. H. Poincare Probab. Statist. 40, no.1, pp53-72] and Lyons et al
(1997)
["A conceptual proof of the Kesten-Stigum theorem for multi-type
branching
processes", Classical and modern branching processes, IMA Vol. Math.
Appl.,
vol.84, Springer, New York, pp181-185], and they should also extend
to more
general classes of branching diffusions. Importantly, the techniques
in this
paper also pave the way for the path large-deviation results for
branching
diffusions found in Hardy and Harris (2006) ["A conceptual approach
to a path
result for branching Brownian motion", Stochastic Processes and their
Applications, doi:10.1016/j.spa.2006.05.010].
http://front.math.ucdavis.edu/math.PR/0611056
---------------------------------------------------------------
4942. MARKOVIANITY AND ERGODICITY FOR A SURFACE GROWTH PDE
D. Bloemker and F. Flandoli and M. Romito
The paper analyses a model in surface growth, where uniqueness of weak
solutions seems to be out of reach. We provide the existence of a weak
martingale solution satisfying energy inequalities and having the Markov
property. Furthermore, under non-degeneracy conditions on the noise, we
establish that any such solution is strong Feller and has a unique
invariant
measure.
http://front.math.ucdavis.edu/math.PR/0611021
---------------------------------------------------------------
4943. ERGODIC PROPERTIES OF GEOMETRICAL CRYSTALLIZATION PROCESSES, I
Y. Davydov (University Lille 1) and A. Illig (University of
Versailles Saint Quentin)
We are interested here in a birth-and-growth process where germs are
born
according to a Poisson point process with invariant under translation
in space
intensity measure. The germs can be born in free space and then start
growing
until occupying the available space. In order to consider various way of
growing, we describe the crystals at each time through their geometrical
properties. In this general framework, the crystallization process
can be
caracterized by the random field giving for a point in the space
state the
first time this point is reached by a crystal. We prove under general
conditions that this random field is mixing in the sens of ergodic
theory and
obtain estimates for the coefficient of absolute regularity.
http://front.math.ucdavis.edu/math.PR/0610966
---------------------------------------------------------------
4944. SECOND LOOK AT THE SPREAD OF EPIDEMICS ON NETWORKS
Eben Kenah and James Robins
In an important paper, M.E.J. Newman claimed that a large class of
network-based stochastic SIR epidemic models are isomorphic to bond
percolation
models, where the bonds are the edges of the contact network and the
bond
occupation probability is equal to the marginal probability of
transmission
from an infected node to a susceptible neighbor. In this paper, we
show that
this isomorphism is incorrect and define a percolation model on a
semi-directed
network that we call the percolation network that is exactly
isomorphic to the
SIR epidemic model. We show that the percolation network model
predicts the
same mean outbreak size (below the epidemic threshold), epidemic
threshold, and
final size of an epidemic as the bond percolation model. However, we
also show
that the bond percolation model fails to predict the correct outbreak
size
distribution and probability of an epidemic for any SIR epidemic
model with a
non-degenerate distribution of infectiousness. In a series of
simulations, we
show that the percolation network model accurately predicts the
probability of
an outbreak of size one and the probability of an epidemic, whereas
the bond
percolation model underestimates the first and overestimates the
latter. In an
appendix, we show that a percolation network model can be defined for
any
time-homogeneous SIR epidemic model. We conclude that percolation
networks are
a very general method of analyzing stochastic SIR epidemic models.
http://front.math.ucdavis.edu/q-bio.QM/0610057
---------------------------------------------------------------
4945. CLT FOR L^{P} MODULI OF CONTINUITY OF GAUSSIAN PROCESSES
Michael B. Marcus and Jay Rosen
Let G=\{G(x),x\in R^1\} be a mean zero Gaussian processes with
stationary
increments and set \si ^2(|x-y|)= E(G(x)-G(y))^2. Let f be a
symmetric function
with Ef(\eta)<\ff, where \eta=N(0,1). When \si^2(s) is concave or when
\si^2(s)=s^r$, $1<r\leq 3/2 we have
\lim_{h\downarrow 0}{\int_a^b f(\frac{G(x+h)-G(x)}{\si (h)}) dx -
(b-a)Ef(\eta)\over \sqrt{\Phi(h,\si(h),f,a,b)}}= N(0,1) in law where
\Phi(h,\si(h),f,a,b) is the variance of the numerator.
This result continues to hold when \si^2(s)=s^r, 3/2<r<2, for certain
functions f, depending on the nature of the coefficients in their
Hermite
polynomial expansion.
The asymptotic behavior of \Phi(h,\si(h),f,a,b) at zero, is
described in a
very large number of cases.
http://front.math.ucdavis.edu/math.PR/0610894
---------------------------------------------------------------
4946. HOW TO DETERMINE THE LAW OF THE NOISE DRIVING A SPDE
H. Gottschalk and B. Smii
We consider a stochastic partial differential equation (SPDE) on a
lattice
\partial_t X=(\Delta-m^2)X-\lambda X^p+\eta where $\eta$ is a
space-time
L\'evy noise. A perturbative (in the sense of formal power series)
strong
solution is given by a tree expansion, whereas the correlation
functions of the
solution are given by a perturbative expansion with coefficients that
are
represented as sums over a certain class of graphs, called Parisi-Wu
graphs.
The perturbative expansion of the truncated (connected) correlation
functions
is obtained via a Linked Cluster Theorem as a sums over connected
graphs only.
The moments of the stationary solution can be calculated as well. In
all these
solutions the cumulants of the single site distribution of the noise
enter as
multiplicative constants. To determine them, e.g. by comparison with a
empirical correlation function, one can fit these constants (e.g. by the
methods of least squares) and thereby one (approximately) determines
law of the
noise.
http://front.math.ucdavis.edu/math.PR/0610906
---------------------------------------------------------------
4947. TWO SUFFICIENT CONDITIONS FOR POISSON APPROXIMATIONS IN THE
FERROMAGNETIC ISING MODEL
David Coupier
A $d$-dimensional ferromagnetic Ising model on a lattice torus is
considered.
As the size of the lattice tends to infinity, two conditions ensuring
a Poisson
approximation for the distribution of the number of occurrences in
the lattice
of any given local configuration are suggested. The proof builds on the
Stein-Chen method. The rate of the Poisson approximation and the
speed of
convergence to it are precised and make sense for the model. Thus,
the two
sufficient conditions are traduced in terms of the magnetic field and
the pair
potential. In particular, the Poisson approximation holds even if both
potentials diverge.
http://front.math.ucdavis.edu/math.PR/0610939
---------------------------------------------------------------
4948. STOCHASTIC MODELS FOR SPECIATION EVENTS IN PHYLOGENETIC TREES
Tanja Gernhard
In a phylogenetic tree, we often don't have information about the time a
speciation event (inner node) occured. Under a neutral model for
speciation, I
develop fast algorithms for calculating the probability that an inner
node i is
the k-th speciation event. For the Yule and the coalescent model, I
develop an
edge length estimation as well. Various properties of the Yule model are
discussed throughout the thesis.
http://front.math.ucdavis.edu/math.CO/0610919
---------------------------------------------------------------
4949. LARGE DEVIATIONS FOR RANDOM MATRIX ENSEMBLES IN MESOSCOPIC PHYSICS
Peter Eichelsbacher and Michael Stolz
In his seminal 1962 paper on the ``threefold way'', Freeman Dyson
classified
the spaces of matrices that support the random matrix ensembles
deemed relevant
from the point of view of classical quantum mechanics. Recently,
Heinzner,
Huckleberry and Zirnbauer have obtained a similar classification
based on less
restrictive assumptions, thus taking care of the needs of modern
mesoscopic
physics. Their list is in one-to-one correspondence with the infinite
families
of Riemannian symmetric spaces as classified by Cartan. The present
paper
develops the corresponding random matrix theories, with a special
emphasis on
large deviation principles.
http://front.math.ucdavis.edu/math.PR/0610811
---------------------------------------------------------------
4950. GRAPH MEASURES
Ilwoo Cho
In this paper, we define several measures induced by a finite
directed graph.
The study themselves is interesting ont only in the noncommutative
probability
point of view but also in the algebraic structure point of view,
since to
define graph measures we defined several rough algebraic structures
induced by
the given graph.
http://front.math.ucdavis.edu/math.PR/0610817
---------------------------------------------------------------
4951. INVERSE PROBLEMS FOR RANDOM WALKS ON TREES: NETWORK TOMOGRAPHY
Victor de la Pena and Henryk Gzyl and Patrick McDonald
Let $G$ be a finite tree with root $r$ and associate to the internal
vertices
of $G$ a collection of transition probabilities for a simple
nondegenerate
Markov chain. Embedd $G$ into a graph $G^\prime$ constructed by
gluing finite
linear chains of length at least 2 to the terminal vertices of $G.$ Then
$G^\prime$ admits distinguished boundary layers and the transition
probabilities associated to the internal vertices of $G$ can be
augmented to
define a simple nondegenerate Markov chain $X$ on the vertices of $G^
\prime.$
We show that the transition probabilities of $X$ can be recovered
from the
joint distribution of first hitting time and first hitting place of $X
$ started
at the root $r$ for the distinguished boundary layers of $G^\prime.$
http://front.math.ucdavis.edu/math.PR/0610821
---------------------------------------------------------------
4952. L-DIVERGENCE CONSISTENCY FOR A DISCRETE PRIOR
M. Grendar
Posterior distribution over a countable set M of continuous data-
sampling
distributions piles up at L-projection of the true distribution r on M,
provided that the L-projection is unique. If there are several L-
projections of
r on M, then the posterior probability splits among them equally.
http://front.math.ucdavis.edu/math.PR/0610824
---------------------------------------------------------------
4953. GENERAL TRIDIAGONAL RANDOM MATRIX MODELS, LIMITING
DISTRIBUTIONS AND FLUCTUATIONS
Ionel Popescu
In this paper we discuss general tridiagonal matrix models which are
natural
extensions of the ones given by Dumitriu and Edelman. We prove here the
convergence of the distribution of the eigenvalues and compute the
limiting
distributions in some particular cases. We also discuss the limit of
fluctuations, which, in a general context, turn out to be Gaussian.
For the
case of several random matrices, we prove the convergence of the
joint moments
and the convergence of the fluctuations to a Gaussian family.
http://front.math.ucdavis.edu/math.PR/0610827
---------------------------------------------------------------
4954. ORDERED RANDOM WALKS
Peter Eichelsbacher and Wolfgang Konig
We construct the conditional version of $k$ independent and identically
distributed random walks on $\R$ given that they stay in strict order
at all
times. This is a generalisation of so-called non-colliding or non-
intersecting
random walks, the discrete variant of Dyson's Brownian motions, which
have been
considered yet only for nearest-neighbor walks on the lattice. Our only
assumptions are moment conditions on the steps and the validity of
the local
central limit theorem. The conditional process is constructed as a Doob
$h$-transform with some positive regular function $V$ that is
strongly related
with the Vandermonde determinant and reduces to that function for
simple random
walk. Furthermore, we prove an invariance principle, i.e., a
functional limit
theorem towards Dyson's Brownian motions, the continuous analogue.
http://front.math.ucdavis.edu/math.PR/0610850
---------------------------------------------------------------
4955. JOHN MICHAEL HAMMERSLEY (1920-2004)
Geoffrey Grimmett and Dominic Welsh
In writing this biographical memoir of John Hammersley, we have tried to
communicate something of the character of the person, and of the
impact of his
scientific achievements across lattice models (for example, percolation,
self-avoiding walks, first-passage percolation, dimer models),
stochastic
processes (including subadditive ergodic theory), Monte Carlo
methods, applied
probability, statistics, and other areas to which he contributed.
There is also
an extended account of his earlier life, taken from autobiographical
notes
written around 1994, together with a list of his published work.
http://front.math.ucdavis.edu/math.PR/0610862
---------------------------------------------------------------
4956. TALAGRAND INEQUALITY FOR THE SEMICIRCULAR LAW AND ENERGY OF
THE EIGENVALUES OF BETA ENSEMBLES
Ionel Popescu
We give a short proof of the free analogue of the Talagrand
inequality for
the transportation cost to the semicircular which was originally
proved by
Biane and Voiculescu. The proof is based on a convexity argument and
is in the
spirit of the original Talagrand's proof. We also discuss the
convergence,
fluctuations and large deviations of the energy of the eigenvalues of
beta
ensembles, which gives also yet another proof of the convergence of the
eigenvalue distribution to the semicircle law.
http://front.math.ucdavis.edu/math.CA/0610826
---------------------------------------------------------------
4957. ESTIMATING THE RELATIVE ORDER OF SPECIATION OR COALESCENCE
EVENTS ON A GIVEN PHYLOGENY
Tanja Gernhard and Daniel Ford and Rutger Vos and Mike Steel
The reconstruction of large phylogenetic trees from data that violates
clocklike evolution (or as a supertree constructed from any m input
trees)
raises a difficult question for biologists - how can one assign
relative dates
to the vertices of the tree? In this paper we investigate this problem,
assuming a uniform distribution on the order of the inner vertices of
the tree
(which includes, but is more general than, the popular Yule
distribution on
trees). We derive fast algorithms for computing the probability that
(i) any
given vertex in the tree was the j--th speciation event (for each j),
and (ii)
any one given vertex is earlier in the tree than a second given
vertex. We show
how the first algorithm can be used to calculate the expected length
of any
given interior edge in any given tree that has been generated under
either a
constant-rate speciation model, or the coalescent model.
http://front.math.ucdavis.edu/math.CO/0610840
---------------------------------------------------------------
4958. FRACTIONAL SPDES DRIVEN BY SPATIALLY CORRELATED NOISE:
EXISTENCE OF THE SOLUTION AND SMOOTHNESS OF ITS DENSITY
Lahcen Boulanba and M'hamed Eddahbi and Mohamed Mellouk
In this paper we study a class of stochastic partial differential
equations
in the whole space $\mathbb{R}^{d}$, with arbitrary dimension $d\geq 1
$, driven
by a Gaussian noise white in time and correlated in space. The
differential
operator is a fractional derivative operator. We show the existence,
uniqueness
and H\"{o}lder's regularity of the solution. Then by means of Malliavin
calculus, we prove that the law of the solution has a smooth density
with
respect to the Lebesgue measure.
http://front.math.ucdavis.edu/math.PR/0610769
---------------------------------------------------------------
4959. A PARADOX FOR ADMISSION CONTROL OF MULTICLASS QUEUEING NETWORK
WITH DIFFERENTIATED SERVICE
Heng-Qing Ye
In this paper, we present counter-intuitive examples for the multiclass
queueing network system. In the system, each station may serve more
than one
job class with differentiated service priority, and each job may require
service sequentially by more than one service station. In our
examples, the
network performance is improved even when more workloads are admitted
for
service.
http://front.math.ucdavis.edu/math.PR/0610784
---------------------------------------------------------------
4960. GIANT COMPONENT AND VACANT SET FOR RANDOM WALK ON A DISCRETE TORUS
Itai Benjamini and Alain-Sol Sznitman
We consider random walk on a discrete torus E of side-length N, in
sufficiently high dimension d. We investigate the percolative
properties of the
vacant set corresponding to the collection of sites which have not
been visited
by the walk up to time uN^d. We show that when u is chosen small, as
N tends to
infinity, there is with overwhelming probability a unique connected
component
in the vacant set which contains segments of length const log N.
Moreover, this
connected component occupies a non-degenerate fraction of the total
number of
sites N^d of E, and any point of E lies within distance an arbitrary
fractional
power of N from this component.
http://front.math.ucdavis.edu/math.PR/0610802
---------------------------------------------------------------
4961. AN APPLICATION OF JACOBI'S ELLIPTIC FUNCTIONS TO ASYMPTOTIC
PROBABILITIES FOR CONFORMAL RESTRICTION MEASURES
Robert O. Bauer
We show that for the conformal restriction measure with exponent $b$
in the
unit disk on hulls $\gamma$ connecting $e^{ix}$ to 1 the probability
of the
event that $\gamma$ avoids the disk of radius $q$ centered at zero
decays like
$\exp(-b\pi x/(1-q))$ if either $b\in[5/8,1]\cup[5/4,\infty)$ and
$x\in(0,\pi]$, or if $b\in(1,5/4)$, $x\in(0,\pi)$, and $bx\le\pi$.
http://front.math.ucdavis.edu/math.PR/0610805
---------------------------------------------------------------
4962. INVARIANCE PRINCIPLES FOR SPATIAL MULTITYPE GALTON-WATSON TREES
Gr\'{e}gory Marc Miermont (LM-Orsay)
We prove that critical multitype Galton-Watson trees converge after
rescaling
to the Brownian continuum random tree, under the hypothesis that the
offspring
distribution has finite covariance matrices. Our study relies on an
ancestral
decomposition for marked multitype trees. We then couple the
genealogical
structure with a spatial motion, whose step distribution may depend
on the
structure of the tree in a local way, and show that the resulting
discrete
spatial trees converge once suitably rescaled to the Brownian snake,
under some
suitable moment assumptions.
http://front.math.ucdavis.edu/math.PR/0610807
---------------------------------------------------------------
4963. STOCHASTIC FLOWS APPROACH TO DUPIRE'S FORMULA
Benjamin Jourdain (CERMICS)
The probabilistic equivalent formulation of Dupire's PDE is the Put-Call
duality equality. In local volatility models including exponential L
\'{e}vy
jumps, we give a direct probabilistic proof for this result based on
stochastic
flows arguments. This approach also enables us to check the
probabilistic
equivalent formulation of various generalizations of Dupire's PDE
recently
obtained by Pironneau by the adjoint equation technique in the case
of complex
options.
http://front.math.ucdavis.edu/math.PR/0610809
---------------------------------------------------------------
4964. ON TRAVERSABLE LENGTH INSIDE SEMI-CYLINDER IN 2D SUPERCRITICAL
BOND PERCOLATION
Nobuaki Sugimine and Masato Takei
We investigate a limit theorem on traversable length inside semi-
cylinder in
the 2-dimensional supercritical Bernoulli bond percolation, which
gives an
extension of Theorem 2 in Grimmett(1981). This type of limit theorems
was
originally studied for the extinction time for the 1-dimensional contact
process on a finite interval in Wagner and Anantharam(2005).
Actually, our main
result Theorem 2.1 is stated under a rather general 2-dimensional bond
percolation setting.
http://front.math.ucdavis.edu/math.PR/0610744
---------------------------------------------------------------
4965. QUADRATIC BSDES DRIVEN BY A CONTINUOUS MARTINGALE AND
APPLICATION TO UTILITY MAXIMIZATION PROBLEM
Marie-Amelie Morlais
In this paper, we study a class of quadratic Backward Stochastic
Differential
Equations (BSDEs) which arises naturally when studying the problem of
utility
maximization with portfolio constraints. We first establish existence
and
uniqueness results for such BSDEs and then, we give an application to
the
utility maximization problem. Three cases of utility functions will be
discussed: the exponential, power and logarithmic ones.
http://front.math.ucdavis.edu/math.PR/0610749
---------------------------------------------------------------
4966. GLOBAL FLUCTUATIONS IN GENERAL BETA DYSON BROWNIAN MOTION
Martin Bender
We consider a system of diffusing particles on the real line in a
quadratic
external potential and with repulsive electrostatic interaction. The
empirical
measure process is known to converge weakly to a deterministic
measure-valued
process as the number of particles tends to infinity. Provided the
initial
fluctuations are small, the rescaled linear statistics of the
empirical measure
process converge in distribution to a Gaussian limit for sufficiently
smooth
test functions. We derive explicit general formulae for the mean and
covariance
in this central limit theorem by analyzing a partial differential
equation
characterizing the limiting fluctuations.
http://front.math.ucdavis.edu/math.PR/0610750
---------------------------------------------------------------
4967. NEW LOWER BOUND ON THE CRITICAL DENSITY IN CONTINUUM PERCOLATION
Zhenning Kong and Edmund M. Yeh
Percolation theory has become a useful tool for the analysis of large
scale
wireless networks. We investigate the fundamental problem of
characterizing the
critical density for Poisson random geometric graphs in continuum
percolation
theory. By using a probabilistic analysis which incorporates the
clustering
effect in random geometric graphs, we develop a new class of lower
bounds for
the critical density. In particular, the lower bound is substantially
improved
to 0.833. This graph theoretical viewpoint provides a new approach
and a deep
insight for the problem.
http://front.math.ucdavis.edu/math.PR/0610751
---------------------------------------------------------------
4968. IT\^{O}'S FORMULA FOR LINEAR FRACTIONAL PDES
Jorge A. Leon (CINVESTAV-Ipn) and Samy Tindel (IECN)
In this paper we introduce a stochastic integral with respect to the
solution
X of the fractional heat equation on [0,1], interpreted as a divergence
operator. This allows to use the techniques of the Malliavin calculus
in order
to establish an It\^{o}-type formula for the process X.
http://front.math.ucdavis.edu/math.PR/0610753
---------------------------------------------------------------
4969. MALLIAVIN CALCULUS FOR INFINITE-DIMENSIONAL SYSTEMS WITH
ADDITIVE NOISE
Yuri Bakhtin and Jonathan C. Mattingly
We consider an infinite-dimensional dynamical system with polynomial
nonlinearity and additive noise given by a finite number of Wiener
processes.
By studying how randomness is spread by the system we develop a
counterpart of
Hormander's classical theory in this setting. We study the
distributions of
finite-dimensional projections of the solutions and give conditions that
provide existence and smoothness of densities of these distributions
with
respect to the Lebesgue measure. We also apply our results to
concrete SPDEs
such as Stochastic Reaction Diffusion Equation and Stochastic 2D
Navier--Stokes
System.
http://front.math.ucdavis.edu/math.PR/0610754
---------------------------------------------------------------
4970. A SIMPLE PROOF OF A RECURRENCE THEOREM FOR RANDOM WALKS IN $\Z^
{2}$
Jean-Marc Derrien (LM-Brest)
In this note, we prove without using Fourier analysis that the symmetric
square integrable random walks in $\Z^{2}$ are recurrent.
http://front.math.ucdavis.edu/math.PR/0610763
---------------------------------------------------------------
4971. A PRACTICAL GUIDE TO MEASURING THE HURST PARAMETER
Richard G. Clegg
This paper describes, in detail, techniques for measuring the Hurst
parameter. Measurements are given on artificial data both in a raw
form and
corrupted in various ways to check the robustness of the tools in
question.
Measurements are also given on real data, both new data sets and well-
studied
data sets. All data and tools used are freely available for download
along with
simple ``recipes'' which any researcher can follow to replicate these
measurements.
http://front.math.ucdavis.edu/math.ST/0610756
---------------------------------------------------------------
4972. A NON-MEASURABLE SET FROM COIN-FLIPS
Alexander E. Holroyd and Terry Soo
In this expository note, we construct a non-measurable set in the
probability
space of coin flips indexed by the integers.
http://front.math.ucdavis.edu/math.PR/0610705
---------------------------------------------------------------
4973. PERCOLATION ON RANDOM JOHNSON-MEHL TESSELLATIONS AND RELATED
MODELS
Bela Bollobas and Oliver Riordan
We make use of the recent proof that the critical probability for
percolation
on random Voronoi tessellations is 1/2 to prove the corresponding
result for
random Johnson-Mehl tessellations, as well as for two-dimensional
slices of
higher dimensional Voronoi tessellations. Surprisingly, the proof is
a little
simpler for these more complicated models.
http://front.math.ucdavis.edu/math.PR/0610716
---------------------------------------------------------------
4974. THE SHIFT, PROPERTIES AND RECOMMENDATIONS FOR PRACTICAL USE
Nicolas Bouleau (LAMM)
Because the stochastic calculus yields rarely random variables with laws
defined by explicit closed formulas, probabilistic numerical
computations are
done most often by simulation. The simulation by the shift, whose
field of
application is as wide as that of Monte Carlo method, is particularly
relevant
when the simulations use, for each sample, a large number of calls to
the
random function. We give here the theoretical features, the
implementation and
the specific advantages of this method.
http://front.math.ucdavis.edu/math.PR/0610729
---------------------------------------------------------------
4975. A LINEAR PROGRAMMING INEQUALITY WITH APPLICATIONS TO
CONCENTRATION OF MEASURE
Leonid Kontorovich
We prove an elementary yet useful inequality bounding the maximal
value of
certain linear programs. This leads directly to a bound on the
martingale
difference for arbitrarily dependent random variables, providing a
generalization of some recent concentration of measure results. The
linear
programming inequality may be of independent interest.
http://front.math.ucdavis.edu/math.FA/0610712
---------------------------------------------------------------
4976. A FINITE DIFFERENCE METHOD FOR PIECEWISE DETERMINISTIC
PROCESSES WITH MEMORY
Mario Annunziato
In this paper the numerical approximation of solutions of Liouville-
Master
Equations for time-dependent distribution functions of Piecewise
Deterministic
Processes with memory is considered. These equations are linear
hyperbolic PDEs
with non-constant coefficients, and boundary conditions that depend on
integrals over the interior of the integration domain. We construct a
finite
difference method of the first order, by a combination of the upwind
method,
for PDEs, and by a direct quadrature, for the boundary condition. We
analyse
convergence of the numerical solution for distribution functions
evolving
towards an equilibrium. Numerical results for two problems, whose
analytical
solutions are known in closed form, illustrate the theoretical finding.
http://front.math.ucdavis.edu/math.NA/0610725
---------------------------------------------------------------
4977. THE SURFACE TENSION NEAR CRITICALITY OF THE 2D-ISING MODEL
R. J. Messikh
For the two dimensional Ising model, we construct the adequate surface
tension near criticality. The latter quantity has been shown to play
a central
role in the study of phase coexistence in a joint limit where the
temperature
approaches the critical point from below and simultaneously the size
of the
system increases fast enough.
http://front.math.ucdavis.edu/math.PR/0610636
---------------------------------------------------------------
4978. THE RENORMALIZATION TRANSFORMATION FOR TWO-TYPE BRANCHING MODELS
Don A. Dawson and Andreas Greven and Frank den Hollander and
Rongfeng Sun and Jan M. Swart
This paper studies countable systems of linearly and hierarchically
interacting diffusions taking values in the positive quadrant. These
systems
arise in population dynamics for two types of individuals migrating
between and
interacting within colonies. Their large-scale space-time behavior
can be
studied by means of a renormalization program. This program, which
has been
carried out successfully in a number of other cases (mostly one-
dimensional),
is based on the construction and the analysis of a nonlinear
renormalization
transformation, acting on the diffusion function for the components
of the
system and connecting the evolution of successive block averages on
successive
time scales. We identify a general class of diffusion functions on
the positive
quadrant for which this renormalization transformation is well-
defined and,
subject to a conjecture on its boundary behavior, can be iterated.
Within
certain subclasses, we identify the fixed points for the
transformation and
investigate their domains of attraction. These domains of attraction
constitute
the universality classes of the system under space-time scaling.
http://front.math.ucdavis.edu/math.PR/0610645
---------------------------------------------------------------
4979. REVISITING OFFSPRING MAXIMA IN BRANCHING PROCESSES
George P. Yanev
We present a progress report for studies on maxima related to
offspring in
branching processes. We summarize and discuss the findings on the
subject that
appeared in the last ten years. Some of the results are refined and
illustrated
with new examples.
http://front.math.ucdavis.edu/math.PR/0610647
---------------------------------------------------------------
4980. ON THE LYAPUNOV EXPONENT OF A MULTIDIMENSIONAL STOCHASTIC FLOW
M. Baldini
Let $X_t$ be a reversible and positive recurrent diffusion in $R^d$
described
by \begin{equation}\nonumber X_t=x+\sigma b(t)+\int_0^tm(X_s)\dif s,
\end{equation} where the diffusion coefficient $\sigma$ is a positive-
definite
matrix and the drift $m$ is a smooth function. Let $X_t(A)$ denote
the image of
a compact set $A\subset R^d$ under the stochastic flow generated by
$X_t$. If
the divergence of the drift is strictly negative, there exists a set of
functions $u$ such that \[\lim_{t\to\infty} \int_{X_t(A)}u(x)\dif
x=0\quad{a.s.} \] A characterization of the functions $u$ is
provided, as well
as lower and upper bounds for the exponential rate of convergence.
http://front.math.ucdavis.edu/math.PR/0610665
---------------------------------------------------------------
4981. GAUSSIAN LIMITS FOR MULTIDIMENSIONAL RANDOM SEQUENTIAL PACKING
AT SATURATION (EXTENDED VERSION)
T. Schreiber and Mathew D. Penrose and J. E. Yukich
Consider the random sequential packing model with infinite input and
in any
dimension. When the input consists of non-zero volume convex solids
we show
that the total number of solids accepted over cubes of volume $\lambda
$ is
asymptotically normal as $\lambda \to \infty$. We provide a rate of
approximation to the normal and show that the finite dimensional
distributions
of the packing measures converge to those of a mean zero generalized
Gaussian
field. The method of proof involves showing that the collection of
accepted
solids satisfies the weak spatial dependence condition known as
stabilization.
http://front.math.ucdavis.edu/math.PR/0610680
---------------------------------------------------------------
4982. CRITICAL EXPONENTS OF PLANAR GRADIENT PERCOLATION
Pierre Nolin (DMA and LM-Orsay)
We study gradient percolation for site percolation on the triangular
lattice.
This is a percolation model where the percolation probability depends
linearly
on the location of the site. We prove the results predicted by
physicists for
this model. More precisely, we describe the fluctuations of the
interfaces
around their (straight) scaling limits, the expected and typical
lengths of
these interfaces. These results build on the recent results for critical
percolation on this lattice by Smirnov, Lawler, Schramm and Werner,
and on the
hyperscaling ideas developed by Kesten.
http://front.math.ucdavis.edu/math.PR/0610682
---------------------------------------------------------------
4983. MONTE CARLO SIMULATIONS WITH GENERALIZED DETAILED BALANCE
USING QUANTUM-CLASSICAL ISOMORPHISM
Yefim I. Leifman
A quantum-classical isomorphism is used in order to develop a Monte
Carlo
simulation with controlled deviation from detailed balance, that is, in
proposed notions, with generalized detailed balance and known
relative entropy
with respect to the reference process at each point. In order to
apply this
method to molecular simulations a new partial chirotope realization
algorithm,
based on linear programming methods, a new distance geometry
algorithm and a
new all-atom off-lattice Monte Carlo method are proposed.
http://front.math.ucdavis.edu/math.PR/0610696
---------------------------------------------------------------
4984. CRITICAL CURVES IN CONFORMALLY INVARIANT STATISTICAL SYSTEMS
I. Rushkin and E. Bettelheim and I. A. Gruzberg and P. Wiegmann
We consider critical curves -- conformally invariant curves that
appear at
critical points of two-dimensional statistical mechanical systems. We
show how
to describe these curves in terms of the Coulomb gas formalism of
conformal
field theory (CFT). We also provide links between this description
and the
stochastic (Schramm-) Loewner evolution (SLE). The connection appears
in the
long-time limit of stochastic evolution of various SLE observables
related to
CFT primary fields. We show how the multifractal spectrum of harmonic
measure
and other fractal characteristics of critical curves can be obtained.
http://front.math.ucdavis.edu/cond-mat/0610550
---------------------------------------------------------------
4985. THE HOLE PROBABILITY FOR GAUSSIAN RANDOM SU(2) POLYNOMIALS
Scott Zrebiec
We show that for Gaussian random SU(2)polynomials of a large degree $N
$ the
probability that there are no zeros in the disk of radius $r$ is less
than
$e^{-c_{1,r} N^2}$, and is also greater than $e^{-c_{2,r} N^2}$.
Enroute to
this result, we also derive a more general result: probability
estimates for
the event that the number of complex zeros of a random polynomial of
high
degree deviates significantly from its mean.
http://front.math.ucdavis.edu/math.CV/0610686
---------------------------------------------------------------
4986. ON SPATIAL THINNING-REPLACEMENT PROCESSES BASED ON VORONOI CELLS
Konstantin Borovkov and David Odell
We introduce a new class of spatial-temporal point processes based on
Voronoi
tessellations. At each step of such a process, a point is chosen at
random
according to a distribution determined by the associated Voronoi
cells. The
point is then removed, and a new random point is added to the
configuration.
The dynamics are simple and intuitive and could be applied to
modeling natural
phenomena. We prove ergodicity of these processes under wide conditions.
http://front.math.ucdavis.edu/math.PR/0610606
---------------------------------------------------------------
4987. STOCHASTIC INTEGRATION IN UMD BANACH SPACES
Jan van Neerven and Mark Veraar and Lutz Weis
In this paper we construct a theory of stochastic integration of
processes
with values in $\calL(H,E)$, where $H$ is a separable Hilbert space
and $E$ is
a UMD Banach space. The integrator is an $H$-cylindrical Brownian
motion. Our
approach is based on a two-sided $L^p$-decoupling inequality for UMD
spaces due
to Garling, which is combined with the theory of stochastic
integration of
$\calL(H,E)$-valued functions introduced recently by two of the
authors. We
obtain various characterizations of the stochastic integral and prove
versions
of the It\^o isometry, the Burkholder-Davis-Gundy inequalities, and the
representation theorem for Brownian martingales.
http://front.math.ucdavis.edu/math.PR/0610619
---------------------------------------------------------------
4988. IDENTIFYING THE DIFFUSION COVARIATION AND THE CO-JUMPS GIVEN
DISCRETE OBSERVATIONS
Fabio Gobbi and Cecilia Mancini
In this paper we consider two processes driven by diffusions and
jumps. We
consider both finite activity and infinite activity jump components.
Given
discrete observations we disentangle the covariation between the two
diffusion
parts from the co-jumps. A commonly used approach to estimate the
diffusion
covariation part is to take the sum of the cross products of the two
processes
increments; however this estimator can be highly biased in the
presence of jump
components, since it approaches the quadratic covariation containing
also the
co-jumps. Our estimator is based on a threshold principle allowing to
isolate
the jumps. %detect the presence of jumps. As a consequence we find an
estimator
which is consistent. In the case of finite activity jump components the
estimator is also asymptotically Gaussian. We assess the performance
of our
estimator for finite samples on four different simulated models.
http://front.math.ucdavis.edu/math.PR/0610621
---------------------------------------------------------------
4989. THE BROWNIAN NET
Rongfeng Sun and Jan M. Swart
The (standard) Brownian web is a collection of coalescing one-
dimensional
Brownian motions, starting from each point in space and time. It
arises as the
diffusive scaling limit of a collection of coalescing random walks.
We show
that it is possible to obtain a nontrivial limiting object if the
random walks
in addition branch with a small probability. We call the limiting
object the
Brownian net, and study some of its elementary properties.
http://front.math.ucdavis.edu/math.PR/0610625
---------------------------------------------------------------
4990. CONSTRUCTION OF A GIBBS MEASURE ASSOCIATED TO THE PERIODIC
BENJAMIN-ONO EQUATION
N. Tzvetkov
We define a finite Borel measure of Gibbs type, supported by the Sobolev
spaces of negative indexes on the circle. The measure can be seen as
a limit of
finite dimensional measures. These finite dimensional measures are
invariant by
the ODE's which correspond to the projection of the Benjamin-Ono
equation,
posed on the circle, on the first N>>1 modes in the trigonometric bases.
http://front.math.ucdavis.edu/math.AP/0610626
---------------------------------------------------------------
4991. UNIQUENESS OF THE CRITICAL PROBABILITY FOR PERCOLATION IN THE
TWO DIMENSIONAL SIERPINSKI CARPET LATTICE
Yasunari Higuchi and Xian-Yuan Wu
We prove that the critical probability for the Sierpinski carpet
lattice in
two dimensions is uniquely determined. The transition is sharp. This
extends
the Kumagai's result to the original Sierpinski carpet lattice.
http://front.math.ucdavis.edu/math.PR/0610583
---------------------------------------------------------------
4992. HOEFFDING DECOMPOSITIONS AND TWO-COLOUR URN SEQUENCES
Omar El-Dakkak (LSTA) and Giovanni Peccati (LSTA)
Let X be a non-deterministic infinite exchangeable sequence with
values in
{0,1}. We show that X is Hoeffding-decomposable if, and only if, X is
either an
i.i.d. sequence or a Polya sequence. This completes the results
established in
Peccati [2004]. The proof uses several combinatorial implications of the
correspondence between Hoeffding decomposability and weak
independence. Our
results must be compared with previous characterizations of i.i.d.
and Polya
sequences given by Hill et al. [1987] and Diaconis and Yilvisaker
[1979].
http://front.math.ucdavis.edu/math.PR/0610590
---------------------------------------------------------------
4993. ON PERMANENTAL PROCESSES
Nathalie Eisenbaum and Haya Kaspi
Permanental processes can be viewed as a generalisation of squared
centered
Gaussian processes. We develop in this paper two main subjects. The
first one
analyses the connections of these processes with the local times of
general
Markov processes. The second deals with Bosonian point processes and the
Bose-Einstein condensation. The obtained results in both directions
are related
and based on the notion of infinite divisibility.
http://front.math.ucdavis.edu/math.PR/0610600
---------------------------------------------------------------
4994. NON-BACKTRACKING RANDOM WALKS MIX FASTER
Noga Alon and Itai Benjamini and Eyal Lubetzky and Sasha Sodin
We compute the mixing rate of a non-backtracking random walk on a
regular
expander. Using some properties of Chebyshev polynomials of the
second kind, we
show that this rate may be up to twice as fast as the mixing rate of
the simple
random walk. The closer the expander is to a Ramanujan graph, the
higher the
ratio between the above two mixing rates is.
As an application, we show that if $G$ is a high-girth regular
expander on
$n$ vertices, then a typical non-backtracking random walk of length $n
$ on $G$
does not visit a vertex more than $(1+o(1))\frac{\log n}{\log\log n}$
times,
and this result is tight. In this sense, the multi-set of visited
vertices is
analogous to the result of throwing $n$ balls to $n$ bins uniformly, in
contrast to the simple random walk on $G$, which almost surely visits
some
vertex $\Omega(\log n)$ times.
http://front.math.ucdavis.edu/math.PR/0610550
---------------------------------------------------------------
4995. INVARIANCE PRINCIPLE, MULTIFRACTIONAL GAUSSIAN PROCESSES AND
LONG-RANGE DEPENDENCE
Serge Cohen and Renaud Marty
This paper is devoted to establish an invariance principle where the
limit
process is a multifractional Gaussian process with a multifractional
function
which takes its values in (1/2, 1). Some properties, as regularity
and local
self-similarity, of this process are studied. Moreover the limit
process is
compared to the multifractional Brownian motion.
http://front.math.ucdavis.edu/math.PR/0610551
---------------------------------------------------------------
4996. ON NUMERICAL INTEGRATION BY THE SHIFT AND APPLICATION TO WIENER
SPACE
Nicolas Bouleau (LAMM)
The aim of this study is to clarify the consequences of recent
theoretical
results for the numerical computation of expectation by the shift
method, and
in particular to yield sufficient criteria for the existence of speed of
convergence of the type `iterated logarithm' in several situations.
We deepen
the case of the Wiener space because it contains many situations
useful in
applications.
http://front.math.ucdavis.edu/math.PR/0610560
---------------------------------------------------------------
4997. PENALIZATIONS OF WALSH BROWNIAN MOTION
Joseph Najnudel
In this paper, we construct a family of probability measures, by
penalizations of a Walsh Brownian motion with a weight dependent on
its value
and its local time at a time t. We prove that this family converges to a
probability measure as t tends to infinity, and we study the
behaviour of this
limit measure.
http://front.math.ucdavis.edu/math.PR/0610564
---------------------------------------------------------------
4998. DYNKIN'S ISOMORPHISM WITHOUT SYMMETRY
Yves Le Jan (LM-Orsay)
The purpose of this note is to extend Dynkin's isomorphim involving
functionals of the occupation field of a symmetric Markov processes
and of the
associated Gaussian field to a suitable class of non symmetric Markov
processes.
http://front.math.ucdavis.edu/math.PR/0610571
---------------------------------------------------------------
4999. ASYMPTOTIC BEHAVIOR OF A BRANCHING POPULATION BEFORE EXTINCTION
Vyacheslav M. Abramov
Under the assumption that the initial population size of a Galton-Watson
branching process increases to infinity, the paper studies asymptotic
behavior
of the population size before extinction. More specifically, we
establish
asymptotic properties of the conditional moments (which are exactly
defined in
the paper).
http://front.math.ucdavis.edu/math.PR/0610506
---------------------------------------------------------------
5000. VISCOELASTICITY AND L\'{E}VY PROCESSES
Nicolas Bouleau (CERMICS)
We show that the linear viscoelastic materials, and more generally the
physical phenomena to which Biot's relaxation theory is relevant, can
be put in
correspondance with the laws of processes with independent
increments. In the
one dimensional case this correspondence is one to one with
subordinators and
gives rise naturally to a conjugation relation on subordinators.
http://front.math.ucdavis.edu/math.PR/0610507
---------------------------------------------------------------
5001. AN EXTENSION TO THE WIENER SPACE OF THE ARBITRARY FUNCTIONS
PRINCIPLE
Nicolas Bouleau (CERMICS)
The arbitrary functions principle says that the fractional part of $nX$
converges stably to an independent random variable uniformly
distributed on the
unit interval, as soon as the random variable $X$ possesses a density
or a
characteristic function vanishing at infinity. We prove a similar
property for
random variables defined on the Wiener space when the stochastic measure
$dB\_s$ is crumpled on itself.
http://front.math.ucdavis.edu/math.PR/0610509
---------------------------------------------------------------
5002. MAXIMAL INEQUALITIES AND A LAW OF THE ITERATED LOGARITHM FOR
NEGATIVELY ASSOCIATED RANDOM FIELDS
Li Xin Zhang
The exponential inequality of the maximum partial sums is a key to
establish
the law of the iterated logarithm of negatively associated random
variables. In
the one-indexed random sequence case, such inequalities for negatively
associated random variables are established by Shao (2000) by using his
comparison theorem between negatively associated and independent random
variables. In the multi-indexed random field case, the comparison
theorem
fails. The purpose of this paper is to establish the Kolmogorov
exponential
inequality as well a moment inequality of the maximum partial sums of a
negatively associated random field via a different method. By using
these
inequalities, the sufficient and necessary condition for the law of the
iterated logarithm of a negatively associated random field to hold is
obtained.
http://front.math.ucdavis.edu/math.PR/0610511
---------------------------------------------------------------
5003. A NOTE ON THE INVARIANCE PRINCIPLE OF THE PRODUCT OF SUMS OF
RANDOM VARIABLES
Li-Xin Zhang and Wei Huang
In literature, the central limit theorems for the product of sums of
various
random variables have studied. The purpose of this note is to show
that this
kind of results are corollary of the invariance principle.
http://front.math.ucdavis.edu/math.PR/0610515
---------------------------------------------------------------
5004. PRECISE RATES IN THE LAW OF THE ITERATED LOGARITHM
Li-Xin Zhang
Let $X$, $X_1$, $X_2$, $...$ be i.i.d. random variables, and let
$S_n=X_1+...
+ X_n$ be the partial sums and $M_n=\max_{k\le n}|S_k|$ be the
maximum partial
sums. We give the sufficient and necessary conditions for a kind of
limit
theorems to hold on the convergence rate of the tail probabilities of
both
$S_n$ and $M_n$. These results are related to the law of the iterated
logarithm. The results of Gut and Spataru (2000) are special cases of
ours.
http://front.math.ucdavis.edu/math.PR/0610519
---------------------------------------------------------------
5005. PRECISE ASYMPTOTICS IN CHUNG'S LAW OF THE ITERATED LOGARITHM
Li-Xin Zhang
This paper gives sufficent and necessary conditions on a kind of limit
results to hold on the precise convergent rate of an infinite series of
probabilities on the Chung type law of the iterated logarithm.
http://front.math.ucdavis.edu/math.PR/0610520
---------------------------------------------------------------
5006. ON THE RATES OF THE OTHER LAW OF THE LOGARITHM
Li-Xin Zhang
By using the strong approximation, this paper establishes several limit
results on the convergent rate of a infinite series of probabilities
on the
other law of iterated logarithm.
http://front.math.ucdavis.edu/math.PR/0610521
---------------------------------------------------------------
5007. A SECOND ORDER SDE FOR THE LANGEVIN PROCESS REFLECTED AT A
COMPLETELY INELASTIC BOUNDARY
Jean Bertoin (PMA and DMA)
It was shown recently that a Langevin process can be reflected at an
energy
absorbing boundary. Here, we establish that the law of this
reflecting process
can be characterized as the unique weak solution to a certain second
order
stochastic differential equation with constraints, which is in sharp
contrast
with a deterministic analog.
http://front.math.ucdavis.edu/math.PR/0610442
---------------------------------------------------------------
5008. THE CAUCHY PROBLEM AND THE MARTINGALE PROBLEM FOR INTEGRO-
DIFFERENTIAL OPERATORS WITH NON-SMOOTH KERNELS
H. Abels M. Kassmann
We consider the linear integro-differential operator $L$ defined by \
[ Lu(x)
=\int_\Rn (u(x+y) - u(x) - 1_{[1,2]}(\alpha) 1_{\{|y|\leq 2\}}(y)y
\cdot \nabla
u(x)) k(x,y) \sd y . \] Here the kernel $k(x,y)$ behaves like
$|y|^{-d-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is H\"older-
continuous
in the first variable, precise definitions are given below. The aim
of this
work is twofold. On one hand, we study the unique solvability of the
Cauchy
problem corresponding to $L$. On the other hand, we study the martingale
problem for $L$. The analytic results obtained for the deterministic
parabolic
equation guarantee that the martingale problem is well-posed. Our
strategy
follows the classical path of Stroock-Varadhan. The assumptions allow
for cases
that have not been dealt with so far.
http://front.math.ucdavis.edu/math.PR/0610445
---------------------------------------------------------------
5009. SHARP PROBABILITY ESTIMATES FOR RANDOM WALKS WITH BARRIERS
Kevin Ford
We give sharp, uniform estimates for the probability that a random
walk of n
steps on the reals avoids a half-line [y,infinity) given that it ends
at the
point x. The estimates hold for general continuous or lattice
distributions
provided the 4th moment is finite.
http://front.math.ucdavis.edu/math.PR/0610450
---------------------------------------------------------------
5010. THE MIXING TIME OF THE GIANT COMPONENT OF A RANDOM GRAPH
Itai Benjamini and Gady Kozma and Nicholas Wormald
We show that the total variation mixing time of the simple random
walk on the
giant component of supercritical Erdos-Renyi graphs is log^2 n. This
statement
was only recently proved, independently, by Fountoulakis and Reed.
Our proof
follows from a structure result for these graphs which is interesting
in its
own right. We show that these graphs are "decorated expanders" - an
expander
glued to graphs whose size has constant expectation and exponential
tail, and
such that each vertex in the expander is glued to no more than a
constant
number of decorations.
http://front.math.ucdavis.edu/math.PR/0610459
---------------------------------------------------------------
5011. COMPONENT SIZES OF THE RANDOM GRAPH OUTSIDE THE SCALING WINDOW
Asaf Nachmias and Yuval Peres
We provide simple proofs describing the behavior of the largest
component of
the Erdos-Renyi random graph $G(n,p)$ outside of the scaling window,
$p={1+\eps(n) \over n}$ where $\eps(n) \to 0$ but $\eps(n)n^{1/3} \to
\infty$.
http://front.math.ucdavis.edu/math.PR/0610466
---------------------------------------------------------------
5012. DIRICHLET FORMS METHODS, AN APPLICATION TO THE PROPAGATION OF
THE ERROR DUE TO THE EULER SCHEME
Nicolas Bouleau (CERMICS)
We present recent advances on Dirichlet forms methods either to extend
financial models beyond the usual stochastic calculus or to study
stochastic
models with less classical tools. In this spirit, we interpret the
asymptotic
error on the solution of an sde due to the Euler scheme in terms of a
Dirichlet
form on the Wiener space, what allows to propagate this error thanks to
functional calculus.
http://front.math.ucdavis.edu/math.PR/0610475
---------------------------------------------------------------
5013. DIFFERENTIAL CALCULUS FOR DIRICHLET FORMS : THE MEASURE-VALUED
GRADIENT PRESERVED BY IMAGE
Nicolas Bouleau (CERMICS)
In order to develop a differential calculus for error propagation we
study
local Dirichlet forms on probability spaces with square field
operator $\Gamma$
-- i.e. error structures -- and we are looking for an object related to
$\Gamma$ which is linear and with a good behaviour by images. For
this we
introduce a new notion called the measure valued gradient which is a
randomized
square root of $\Gamma$. The exposition begins with inspecting some
natural
notions candidate to solve the problem before proposing the measure-
valued
gradient and proving its satisfactory properties.
http://front.math.ucdavis.edu/math.PR/0610485
---------------------------------------------------------------
5014. DIRICHLET FORMS IN SIMULATION
Nicolas Bouleau (CERMICS)
Equipping the probability space with a local Dirichlet form with
square field
operator $\Gamma$ and generator $A$ allows to improve Monte Carlo
computations
of expectations, densities, and conditional expectations, as soon as
we are
able to simulate a random variable $X$ together with $\Gamma[X]$ and
$A[X]$. We
give examples on the Wiener space, on the Poisson space and on the
Monte Carlo
space. When $X$ is real-valued we give an explicit formula yielding
the density
at the speed of the law of large numbers.
http://front.math.ucdavis.edu/math.PR/0610486
---------------------------------------------------------------
5015. ERROR CALCULUS AND PATH SENSITIVITY IN FINANCIAL MODELS
Nicolas Bouleau (CERMICS)
In the framework of risk management, for the study of the sensitivity of
pricing and hedging in stochastic financial models to changes of
parameters and
to perturbations of the stock prices, we propose an error calculus
which is an
extension of the Malliavin calculus based on Dirichlet forms.
Although useful
also in physics, this error calculus is well adapted to stochastic
analysis and
seems to be the best practicable in finance. This tool is explained here
intuitively and with some simple examples.
http://front.math.ucdavis.edu/math.PR/0610489
---------------------------------------------------------------
5016. CALCUL D'ERREUR COMPLET LIPSCHITZIEN ET FORMES DE DIRICHLET
Nicolas Bouleau (CERMICS)
We study the error calculus from a mathematical point of view, in
particular
for the infinite dimensional models met in stochastic analysis. Gauss
was the
first to propose an error calculus. It can be reinforced by an extension
principle based on Dirichlet forms which gives more strength to the
coherence
property. One gets a Lipschitzian complete error calculus which
behaves well by
images and by products and allows a quick and easy construction of
the basic
mathematical tools of Malliavin calculus. This allows also to revisit
the
delicate question of error permanency that Poincar\'{e} emphasized.
This error
calculus is connected with statistics by mean of the notion of Fisher
information.
http://front.math.ucdavis.edu/math.PR/0610491
---------------------------------------------------------------
5017. NON-EXPONENTIAL STABILITY AND DECAY RATES IN NONLINEAR
STOCHASTIC DIFFERENCE EQUATION WITH UNBOUNDED NOISES
J.A.D. Appleby and G. Berkolaiko and A. Rodkina
We consider stochastic difference equation
$$ x_{n+1} = x_n (1 - h f(x_n) + \sqrt{h} g(x_n) \xi_{n+1}), $$ where
functions f and g are nonlinear and bounded, random variables \xi_i are
independent and h>0 is a nonrandom parameter. We establish results on
asymptotic stability and instability of the trivial solution x_n=0.
We also
show, that for some natural choices of the nonlinearities f and g,
the rate of
decay of x_n is approximately polynomial: we find \alpha>0 such that
x_n decay
faster than n^{-\alpha+\epsilon} but slower than n^{-\alpha-\epsilon}
for any
\epsilon>0. It also turns out that if g(x) decays faster than f(x) as
x->0, the
polynomial rate of decay can be established exactly, x_n n^\alpha ->
const. On
the other hand, if the coefficient by the noise does not decay fast
enough, the
approximate decay rate is the best possible result.
http://front.math.ucdavis.edu/math.PR/0610425
---------------------------------------------------------------
5018. METRIC AND MIXING SUFFICIENT CONDITIONS FOR CONCENTRATION OF
MEASURE
Leonid Kontorovich
We derive sufficient conditions for a family $(X^n,\rho_n,P_n)$ of
metric
probability spaces to have the measure concentration property.
Specifically, if
the sequence $\{P_n\}$ of probability measures satisfies a strong mixing
condition (which we call $\eta$-mixing) and the sequence of metrics
$\{\rho_n\}$ is what we call $\Psi$-dominated, we show that $(X^n,
\rho_n,P_n)$
is a normal Levy family. We establish these properties for some metric
probability spaces, including the possibly novel $X=[0,1]$, $\rho_n=
\ell_1$
case.
http://front.math.ucdavis.edu/math.PR/0610427
---------------------------------------------------------------
5019. THE ORNSTEIN UHLENBECK BRIDGE AND APPLICATIONS TO MARKOV
SEMIGROUPS
Beniamin Goldys and Bohdan Maslowski
For an arbitrary Hilbert space-valued Ornstein-Uhlenbeck process we
construct
the Ornstein-Uhlenbeck Bridge connecting a starting point $x$ and an
endpoint
$y$ that belongs to a certain linear subspace of full measure. We
derive also a
stochastic evolution equation satisfied by the OU Bridge and study
its basic
properties. The OU Bridge is then used to investigate the Markov
transition
semigroup associated to a nonlinear stochastic evolution equation
with additive
noise. We provide an explicit formula for the transition density and
study its
regularity. Given the Strong Feller property and the existence of an
invariant
measure we show that the transition semigroup maps $L^p$ functions into
continuous functions. We also show that transition operators are $q$-
summing
for some $q>p>1$, in particular of Hilbert-Schmidt type.
http://front.math.ucdavis.edu/math.PR/0610386
---------------------------------------------------------------
5020. TH\'{E}OR\`{E}ME DE DONSKER ET FORMES DE DIRICHLET
Nicolas Bouleau (CERMICS)
We use the language of errors to handle local Dirichlet forms with
square
field operator (cf [2]). Let us consider, under the hypotheses of
Donsker
theorem, a random walk converging weakly to a Brownian motion. If in
addition
the random walk is supposed to be erroneous, the convergence occurs
in the
sense of Dirichlet forms and induces the Ornstein-Uhlenbeck structure
on the
Wiener space. This quite natural result uses an extension of Donsker
theorem to
functions with quadratic growth. As an application we prove an
invariance
principle for the gradient of the maximum of the Brownian path
computed by
Nualart and Vives.
http://front.math.ucdavis.edu/math.PR/0610392
---------------------------------------------------------------
5021. SUBMEAN VARIANCE BOUND FOR EFFECTIVE RESISTANCE ON RANDOM
ELECTRIC NETWORKS
Itai Benjamini and Raphael Rossignol
We study a model of random electric networks with Bernoulli
resistances. In
the case of the lattice Z^2, we show that the point-to-point effective
resistance has a small variance compared to its expected value,
whereas for
Z^d, with d different from 2, expectation and variance are of the
same order.
Similar results are obtained in the context of p-resistance. The
proofs rely on
a modified Poincare inequality due to Falik and Samorodnitsky.
http://front.math.ucdavis.edu/math.PR/0610393
---------------------------------------------------------------
5022. THE CURIE-WEISS MODEL WITH DYNAMICAL EXTERNAL FIELD
Clement Dombry and Nadine Guillotin-Plantard
We study a Curie-Weiss model with a random external field generated by a
dynamical system. Probabilistic limit theorems (weak law of large
numbers,
central limit theorems) are proven for the corresponding magnetization.
http://front.math.ucdavis.edu/math.PR/0610394
---------------------------------------------------------------
5023. A STOCHASTIC APPROXIMATION SCHEME AND CONVERGENCE THEOREM FOR
PARTICLE INTERACTIONS WITH PERFECTLY REFLECTING BOUNDARIES
Clive G. Wells
We prove the existence of a solution to an equation governing the number
density within a compact domain of a discrete particle system for a
prescribed
class of particle interactions taking into account the effects of the
diffusion
and drift of the set of particles. Each particle carries a number of
internal
coordinates which may evolve continuously in time, determined by what
we will
refer to as the internal drift, or discretely via the interaction
kernels.
Perfectly reflecting boundary conditions are imposed on the system
and all the
processes may be spatially and temporally inhomogeneous. We use a
relative
compactness argument to construct a sequence of measures that
converge weakly
to a solution of the governing equation. Since the proof of existence
is a
constructive one, it provides a stochastic approximation scheme that
can be
used for the numerical study of molecular dynamics.
http://front.math.ucdavis.edu/math.PR/0610412
---------------------------------------------------------------
5024. WHEN AND HOW AN ERROR YIELDS A DIRICHLET FORM
Nicolas Bouleau (CERMICS)
We consider a random variable $Y$ and approximations $Y\_n$, defined
on the
same probability space with values in the same measurable space as $Y
$. We are
interested in situations where the approximations $Y\_n$ allow to
define a
Dirichlet form in the space $L^2(P\_Y)$ where $P\_Y$ is the law of $Y
$. Our
approach consists in studying both biases and variances. The article
attempts
to propose a general theoretical framework. It is illustrated by several
examples.
http://front.math.ucdavis.edu/math.FA/0610389
---------------------------------------------------------------
5025. SOME THOUGHTS UPON AXIOMATIZED LANGUAGES WITH ESTENSION TOOLS,
A FOCUS ON PROBABILITY THEORY AND ERROR CALCULUS WITH DIRICHLET FORMS
Nicolas Bouleau (CERMICS)
A comparison of the "theory of random sequences" developed during the
twentieth century and the axiomatic approach of probability theory
proposed by
Kolmogorov shows the importance of sigma-additivity as extension tool.
Similarly, the Cauchy criterion appears to be an extension tool for
mathematical analysis. The Dirichlet forms theory possesses also such an
extension tool. They are the source of the fruitfulness of these
languages and
the condition of their creativity. A connection is given with the so-
called
Richard paradox.
http://front.math.ucdavis.edu/math.HO/0610390
---------------------------------------------------------------
5026. A TIGHT BOUND FOR THE LAMPLIGHTER PROBLEM
Murali K. Ganapathy and Prasad Tetali
We settle an open problem, raised by Y. Peres and D. Revelle,
concerning the
$L^2$ mixing time of the random walk on the lamplighter graph. We
also provide
general bounds relating the entropy decay of a Markov chain to the
separation
distance of the chain, and show that the lamplighter graphs once
again provide
examples of tightness of our results.
http://front.math.ucdavis.edu/math.PR/0610345
---------------------------------------------------------------
5027. CORRECTION. BROWNIAN MODELS OF OPEN PROCESSING NETWORKS:
CANONICAL REPRESENTATION OF WORKLOAD
J. Michael Harrison
Due to a printing error the above mentioned article [Annals of Applied
Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous
equations appearing incorrectly in the print version of this paper.
The entire
article follows as it should have appeared. IMS apologizes to the
author and
the readers for this error. A recent paper by Harrison and Van Mieghem
explained in general mathematical terms how one forms an ``equivalent
workload
formulation'' of a Brownian network model. Denoting by $Z(t)$ the
state vector
of the original Brownian network, one has a lower dimensional state
descriptor
$W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be
chosen as
any basis matrix for a particular linear space. This paper considers
Brownian
models for a very general class of open processing networks, and in that
context develops a more extensive interpretation of the equivalent
workload
formulation, thus extending earlier work by Laws on alternate routing
problems.
A linear program called the static planning problem is introduced to
articulate
the notion of ``heavy traffic'' for a general open network, and the
dual of
that linear program is used to define a canonical choice of the basis
matrix
$M$. To be specific, rows of the canonical $M$ are alternative basic
optimal
solutions of the dual linear program. If the network data satisfy a
natural
monotonicity condition, the canonical matrix $M$ is shown to be
nonnegative,
and another natural condition is identified which ensures that $M$
admits a
factorization related to the notion of resource pooling.
http://front.math.ucdavis.edu/math.PR/0610352
---------------------------------------------------------------
5028. ON SOME ERRORS RELATED TO THE GRADUATION OF MEASURINF INSTRUMENTS
Nicolas Bouleau (CERMICS)
The error on a real quantity Y due to the graduation of the measuring
instrument may be represented, when the graduation is regular and
fines down,
by a Dirichlet form on R whose square field operator do not depend on
the
probability law of Y as soon as this law possesses a continuous
density. This
feature is related to the "arbitrary functions principle" (Poincar
\'{e}, Hopf).
We give extensions of this property to multivariate case and infinite
dimensional case for approximations of the Brownian motion. We use a
Girsanov
theorem for Dirichlet forms which has its own interest. Connections
are given
with discretization of stochastic differential equations.
http://front.math.ucdavis.edu/math.PR/0610355
---------------------------------------------------------------
5029. REFLECTION POSITIVITY AND PHASE TRANSITIONS IN LATTICE SPIN MODELS
Marek Biskup
Reflection positivity (RP) is a property of Gibbs measures exhibited
by a
class of lattice spin systems that includes the Ising, Potts and
Heisenberg
models. The RP property is useful because of its two basic consequences:
infrared bound and chessboard estimates. These are one of basic (and
rather
efficient) tools for proving phase transitions in many models of
physical
interest. The content of the notes presented hereby are the lectures on
reflection positivity and its consequences that the author delivered
at the
Prague Summer School on Mathematical Statistical Mechanics in
September 2006.
The notes summarize both the classical material on the subject from
the late
1970s as well as some of the more recent developments.
http://front.math.ucdavis.edu/math-ph/0610025
---------------------------------------------------------------
5030. ASYMPTOTIC FEYNMAN-KAC FORMULAE FOR LARGE SYMMETRISED SYSTEMS
OF RANDOM WALKS
Stefan Adams and Tony Dorlas
We study large deviations principles for $ N $ random processes on the
lattice $ \Z^d $ with finite time horizon $ [0,\beta] $ under a
symmetrised
measure where all initial and terminal points are uniformly given by
a random
permutation. That is, given a permutation $ \sigma $ of $ N $
elements and a
vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random
processes
terminate in the points $ (x_{\sigma(1)},...,x_{\sigma(N)}) $ and
then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-
level
large deviations principles for the mean of empirical path measures,
for the
mean of paths and for the mean of occupation local times under this
symmetrised
measure. The symmetrised measure cannot be written as any product of
single
random process distributions. We show a couple of important
applications of
these results in quantum statistical mechanics using the Feynman-Kac
formulae
representing traces of certain trace class operators. In particular
we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-
Einstein
statistics and mean field interactions.
A special case of our large deviations principle for the mean of
occupation
local times of $ N $ simple random walks has the Donsker-Varadhan
rate function
as the rate function for the limit $ N\to\infty $ but for finite time
$ \beta
$. We give an interpretation in quantum statistical mechanics for this
surprising result.
http://front.math.ucdavis.edu/math-ph/0610026
---------------------------------------------------------------
5031. THE SCALING LIMITS OF PLANAR LERW IN FINITELY CONNECTED DOMAINS
Dapeng Zhan
We define a family of SLE-type processes in finitely connected
domains, which
are called continuous LERW (loop-erased random walk). A continuous LERW
describes a random curve in a finitely connected domain that starts
from a
prime end and ends at a certain target set, which could be an
interior point,
or a prime end, or a side arc. It is defined using the usual chordal
Loewner
equation with the driving function being $\sqrt 2 B(t)$ plus a drift
term. The
distributions of continuous LERW are conformally invariant. A
continuous LERW
preserves a family of local martingales, which are composed of
generalized
Poisson kernels, normalized by their behaviors near the target set.
These local
martingales resemble the discrete martingales preserved by the
corresponding
LERW on the discrete approximation of the domain. For all kinds of
targets, if
the domain satisfies certain boundary conditions, we use these
martingales to
prove that when the mesh of the discrete approximation is small
enough, the
continuous LERW and the corresponding discrete LERW can be coupled
together,
such that after a suitable parametrization, with probability close to
1, the
two curves are uniformly close to each other.
http://front.math.ucdavis.edu/math.PR/0610304
---------------------------------------------------------------
5032. ON THE ERGODICITY PROPERTIES OF SOME ADAPTIVE MCMC ALGORITHMS
Christophe Andrieu and \'{E}ric Moulines
In this paper we study the ergodicity properties of some adaptive Markov
chain Monte Carlo algorithms (MCMC) that have been recently proposed
in the
literature. We prove that under a set of verifiable conditions, ergodic
averages calculated from the output of a so-called adaptive MCMC sampler
converge to the required value and can even, under more stringent
assumptions,
satisfy a central limit theorem. We prove that the conditions
required are
satisfied for the independent Metropolis--Hastings algorithm and the
random
walk Metropolis algorithm with symmetric increments. Finally, we
propose an
application of these results to the case where the proposal
distribution of the
Metropolis--Hastings update is a mixture of distributions from a curved
exponential family.
http://front.math.ucdavis.edu/math.PR/0610317
---------------------------------------------------------------
5033. NONMONOTONICITY OF PHASE TRANSITIONS IN A LOSS NETWORK WITH
CONTROLS
Brad Luen and Kavita Ramanan and Ilze Ziedins
We consider a symmetric tree loss network that supports single-link
(unicast)
and multi-link (multicast) calls to nearest neighbors and has
capacity $C$ on
each link. The network operates a control so that the number of
multicast calls
centered at any node cannot exceed $C_V$ and the number of unicast
calls at a
link cannot exceed $C_E$, where $C_E$, $C_V\leq C$. We show that
uniqueness of
Gibbs measures on the infinite tree is equivalent to the convergence
of certain
recursions of a related map. For the case $C_V=1$ and $C_E=C$, we
precisely
characterize the phase transition surface and show that the phase
transition is
always nonmonotone in the arrival rate of the multicast calls. This
model is an
example of a system with hard constraints that has weights attached
to both the
edges and nodes of the network and can be viewed as a generalization
of the
hard core model that arises in statistical mechanics and
combinatorics. Some of
the results obtained also hold for more general models than just the
loss
network. The proofs rely on a combination of techniques from
probability theory
and dynamical systems.
http://front.math.ucdavis.edu/math.PR/0610321
---------------------------------------------------------------
5034. ON THE VARIATIONAL DISTANCE OF TWO TREES
M. A. Steel and L. A. Sz\'{e}kely
A widely studied model for generating sequences is to ``evolve'' them
on a
tree according to a symmetric Markov process. We prove that model
trees tend to
be maximally ``far apart'' in terms of variational distance.
http://front.math.ucdavis.edu/math.PR/0610323
---------------------------------------------------------------
5035. ON THE VALUE OF OPTIMAL STOPPING GAMES
Erik Ekstr\"{o}m and Stephane Villeneuve
We show, under weaker assumptions than in the previous literature,
that a
perpetual optimal stopping game always has a value. We also show that
there
exists an optimal stopping time for the seller, but not necessarily
for the
buyer. Moreover, conditions are provided under which the existence of an
optimal stopping time for the buyer is guaranteed. The results are
illustrated
explicitly in two examples.
http://front.math.ucdavis.edu/math.PR/0610324
---------------------------------------------------------------
5036. UPPER LIMITS OF SINAI'S WALK IN RANDOM SCENERY
Olivier Zindy (PMA)
We consider Sinai's walk in i.i.d. random scenery and focus our
attention on
a conjecture of R\'ev\'esz \cite{r05} concerning the upper limits of
Sinai's
walk in random scenery when the scenery is bounded from above. A
close study of
the competition between the concentration property for Sinai's walk and
negative values for the scenery enables us to prove that the
conjecture is true
if the scenery has "thin" negative tails and is false otherwise.
http://front.math.ucdavis.edu/math.PR/0610326
---------------------------------------------------------------
5037. A HETEROPOLYMER IN A MEDIUM WITH RANDOM DROPLETS
Mario V. W\"{u}thrich
We define a heteropolymer in a medium with random droplets. We prove
that for
this model we have two regimes: a delocalized one and a localized
one. In the
localized regime we prove tightness to the droplets, whereas in the
delocalized
regime we prove diffusive path behavior.
http://front.math.ucdavis.edu/math.PR/0610328
---------------------------------------------------------------
5038. CONVERGENCE RATE AND AVERAGING OF NONLINEAR TWO-TIME-SCALE
STOCHASTIC APPROXIMATION ALGORITHMS
Abdelkader Mokkadem and Mariane Pelletier
The first aim of this paper is to establish the weak convergence rate of
nonlinear two-time-scale stochastic approximation algorithms. Its
second aim is
to introduce the averaging principle in the context of two-time-scale
stochastic approximation algorithms. We first define the notion of
asymptotic
efficiency in this framework, then introduce the averaged two-time-scale
stochastic approximation algorithm, and finally establish its weak
convergence
rate. We show, in particular, that both components of the averaged
two-time-scale stochastic approximation algorithm simultaneously
converge at
the optimal rate $\sqrt{n}$.
http://front.math.ucdavis.edu/math.PR/0610329
---------------------------------------------------------------
5039. TRANSLATION-INVARIANT MATCHINGS OF COIN-FLIPS ON Z^D
Terry Soo
Consider independent fair coin flips at each site of the lattice Z^d. We
study non-randomized translation-invariant perfect matching rules of
occupied
sites to unoccupied sites and determine bounds on the distance from a
site to
its partner. In particular, in d=2, if Z is the distance from the
origin to its
partner then we obtain that if 0 < p < 2/3, then the p-th moment of Z is
finite. This is related to an open problem of Holroyd and Peres.
http://front.math.ucdavis.edu/math.PR/0610334
---------------------------------------------------------------
5040. ASYMPTOTICS OF PLANCHEREL-TYPE RANDOM PARTITIONS
Alexei Borodin and Grigori Olshanski
We present a solution to a problem suggested by Philippe Biane: We
prove that
a certain Plancherel-type probability distribution on partitions
converges, as
partitions get large, to a new determinantal random point process on
the set
{0,1,2,...} of nonnegative integers. This can be viewed as an edge limit
ransition. The limit process is determined by a correlation kernel on
{0,1,2,...} which is expressed through the Hermite polynomials, we
call it the
discrete Hermite kernel. The proof is based on a simple argument
which derives
convergence of correlation kernels from convergence of unbounded self-
adjoint
difference operators.
Our approach can also be applied to a number of other
probabilistic models.
As an example, we discuss a bulk limit for one more Plancherel-type
model of
random partitions.
http://front.math.ucdavis.edu/math.PR/0610240
---------------------------------------------------------------
5041. STOCHASTIC VOLTERRA EQUATIONS DRIVEN BY CYLINDRICAL WIENER PROCESS
Anna Karczewska and Carlos Lizama
In this paper, stochastic Volterra equations driven by cylindrical and
genuine Wiener process in Hilbert space are investigated. Sufficient
conditions
for existence of strong solutions are given. The key role is played by
convergence of $\alpha$-times resolvent families.
http://front.math.ucdavis.edu/math.PR/0610241
---------------------------------------------------------------
5042. MARTIN BOUNDARY OF A REFLECTED RANDOM WALK ON A HALF-SPACE
Irina Ignatiouk-Robert
The complete representation of the Martin compactification for reflected
random walks on a half-space $\Z^d\times\N$ is obtained. It is shown
that the
full Martin compactification is in general not homeomorphic to the
``radial''
compactification obtained by Ney and Spitzer for the homogeneous
random walks
in $\Z^d$ : convergence of a sequence of points $z_n\in\Z^{d-1}\times
\N$ to a
point of on the Martin boundary does not imply convergence of the
sequence
$z_n/|z_n|$ on the unit sphere $S^d$. Our approach relies on the large
deviation properties of the scaled processes and uses Pascal's method
combined
with the ratio limit theorem. The existence of non-radial limits is
related to
non-linear optimal large deviation trajectories.
http://front.math.ucdavis.edu/math.PR/0610242
---------------------------------------------------------------
5043. THE PALM MEASURE AND THE VORONOI TESSELLATION FOR THE GINIBRE
PROCESS
Andre Goldman
We prove that the Palm measure of the Ginibre process is obtained by
removing
a gaussian distributed point from the process and adding the origin.
We obtain
also precise formulas describing the law of the typical cell of the
Ginibre-Voronoi tessellation. We show that near the cell's germs a more
important part of the area is captured in the Ginibre-Voronoi
tessellation than
in the case of the Poisson-Voronoi tessellation. Moments areas of
corresponding
subdomains of cells are explicitly evaluated.
http://front.math.ucdavis.edu/math.PR/0610243
---------------------------------------------------------------
5044. ON STOCHASTIC FRACTIONAL VOLTERRA EQUATIONS IN HILBERT SPACE
Anna Karczewska and Carlos Lizama
In this paper stochastic Volterra equations admitting exponentially
bounded
resolvents are studied. After obtaining convergence of resolvents, some
properties of stochastic convolutions are given. The paper provides a
sufficient condition for a stochastic convolution to be a strong
solution to a
stochastic Volterra equation.
http://front.math.ucdavis.edu/math.PR/0610244
---------------------------------------------------------------
5045. TAIL ASYMPTOTICS FOR THE MAXIMUM OF PERTURBED RANDOM WALK
Victor F. Araman and Peter W. Glynn
Consider a random walk $S=(S_n:n\geq 0)$ that is ``perturbed'' by a
stationary sequence $(\xi_n:n\geq 0)$ to produce the process
$(S_n+\xi_n:n\geq0)$. This paper is concerned with computing the
distribution
of the all-time maximum $M_{\infty}=\max \{S_k+\xi_k:k\geq0\}$ of
perturbed
random walk with a negative drift. Such a maximum arises in several
different
applications settings, including production systems, communications
networks
and insurance risk. Our main results describe asymptotics for
$\mathbb{P}(M_{\infty}>x)$ as $x\to\infty$. The tail asymptotics
depend greatly
on whether the $\xi_n$'s are light-tailed or heavy-tailed. In the
light-tailed
setting, the tail asymptotic is closely related to the Cram\'{e}r--
Lundberg
asymptotic for standard random walk.
http://front.math.ucdavis.edu/math.PR/0610271
---------------------------------------------------------------
5046. RANDOM REWARDS, FRACTIONAL BROWNIAN LOCAL TIMES AND STABLE SELF-
SIMILAR PROCESSES
Serge Cohen and Gennady Samorodnitsky
We describe a new class of self-similar symmetric $\alpha$-stable
processes
with stationary increments arising as a large time scale limit in a
situation
where many users are earning random rewards or incurring random
costs. The
resulting models are different from the ones studied earlier both in
their
memory properties and smoothness of the sample paths.
http://front.math.ucdavis.edu/math.PR/0610272
---------------------------------------------------------------
5047. REPRESENTATIONS OF LIE GROUPS AND RANDOM MATRICES
Benoit Collins and Piotr Sniady
We study the asymptotics of representations of a fixed compact Lie
group. We
prove that the limit behavior of a sequence of such representations
can be
described in terms of certain random matrices; in particular
operations on
representations (for example: tensor product, restriction to a subgroup)
correspond to some natural operations on random matrices
(respectively: sum of
independent random matrices, taking the corners of a random matrix).
Our method
of proof is to treat the canonical block matrix associated to a
representation
as a random matrix with non-commutative entries.
http://front.math.ucdavis.edu/math.PR/0610285
---------------------------------------------------------------
5048. A COMPLETE RENORMALIZATION GROUP TRAJECTORY BETWEEN TWO FIXED
POINTS
Abdelmalek Abdesselam
We give a rigorous nonperturbative construction of a massless discrete
trajectory for Wilson's exact renormalization group. The model is a
three
dimensional Euclidean field theory with a modified free propagator. The
trajectory realizes the mean field to critical crossover from the
ultraviolet
Gaussian fixed point to an analog recently constructed by Brydges,
Mitter and
Scoppola of the Wilson-Fisher nontrivial fixed point.
http://front.math.ucdavis.edu/math-ph/0610018
---------------------------------------------------------------
5049. MULTIVARIABLE CHRISTOFFEL-DARBOUX KERNELS AND CHARACTERISTIC
POLYNOMIALS OF RANDOM HERMITIAN MATRICES
Hjalmar Rosengren
We study multivariable Christoffel-Darboux kernels, which may be
viewed as
reproducing kernels for antisymmetric orthogonal polynomials, and
also as
correlation functions for products of characteristic polynomials of
random
hermitian matrices. Using their interpretation as reproducing
kernels, we
obtain simple proofs of pfaffian and determinant formulas, as well as
Schur
polynomial expansions, for such kernels. In subsequent work, these
results are
applied in combinatorics (enumeration of marked shifted tableaux) and
number
theory (representation of integers as sums of squares).
http://front.math.ucdavis.edu/math.CA/0606391
---------------------------------------------------------------
5050. SPECTRAL GAP FOR STABLE PROCESS ON CONVEX PLANAR DOUBLE
SYMMETRIC DOMAINS
Bartlomiej Dyda and Tadeusz Kulczycki
We study the semigroup of the symmetric $\alpha$-stable process in
bounded
domains in $\R^2$. We obtain a variational formula for the spectral
gap, i.e.
the difference between two first eigenvalues of the generator of this
semigroup. This variational formula allows us to obtain lower bound
estimates
of the spectral gap for convex planar domains which are symmetric
with respect
to both coordinate axes. For rectangles, using "midconcavity" of the
first
eigenfunction, we obtain sharp upper and lower bound estimates of the
spectral
gap.
http://front.math.ucdavis.edu/math.SP/0610283
---------------------------------------------------------------
5051. THE NUMBER OF UNBOUNDED COMPONENTS IN THE POISSON-BOOLEAN MODEL
ON THE HYPERBOLIC DISC
Johan Tykesson
We consider the Poisson-Boolean continuum percolation model on the
hyperbolic
disc. We show that there are intensities for the underlying Poisson
point
process for which there are infinitely many unbounded connected
components in
the covered and vacant regions of the hyperolic disc.
http://front.math.ucdavis.edu/math.PR/0610202
---------------------------------------------------------------
5052. GAMMA TILTING CALCULUS FOR GGC AND DIRICHLET MEANS WITH
APPLICATIONS TO LINNIK PROCESSES AND OCCUPATION TIME LAWS FOR
RANDOMLY SKEWED BESSEL
PROCESSES AND BRIDGES
Lancelot F. James
This paper develops some general calculus for GGC and Dirichlet
process means
functionals. It then proceeds via an investigation of positive Linnik
random
variables, and more generally random variables derived from
compositions of a
stable subordinator with GGC subordinators, to establish various
distributional
equivalences between these models and phenomena connected to local
times and
occupation times of what are defined as randomly skewed Bessel
processes and
bridges. This yields a host of interesting identities and explicit
density
formula for these models. Randomly skewed Bessel processes and
bridges may be
seen as a randomization of their p-skewed counterparts developed in
Barlow,
Pitman and Yor~(1989) and Pitman and Yor~(1997), and are shown to
naturally
arise via exponential tilting. As a special result it is shown that the
occupation time of a p-skewed random Bessel process or (generalized)
bridge is
equivalent in distribution to the the occupation time of a non-
trivial randomly
skewed process.
http://front.math.ucdavis.edu/math.PR/0610218
---------------------------------------------------------------
5053. THE MINIMAL ENTROPY MARTINGALE MEASURE FOR GENERAL BARNDORFF-
NIELSEN/SHEPHARD MODELS
Thorsten Rheinl\"{a}nder and Gallus Steiger
We determine the minimal entropy martingale measure for a general
class of
stochastic volatility models where both price process and volatility
process
contain jump terms which are correlated. This generalizes previous
studies
which have treated either the geometric L\'{e}vy case or continuous
price
processes with an orthogonal volatility process. We proceed by
linking the
entropy measure to a certain semi-linear integro-PDE for which we
prove the
existence of a classical solution.
http://front.math.ucdavis.edu/math.PR/0610219
---------------------------------------------------------------
5054. ON THE TWO-TIMES DIFFERENTIABILITY OF THE VALUE FUNCTIONS IN
THE PROBLEM OF OPTIMAL INVESTMENT IN INCOMPLETE MARKETS
Dmitry Kramkov and Mihai S\^{{\i}}rbu
We study the two-times differentiability of the value functions of
the primal
and dual optimization problems that appear in the setting of expected
utility
maximization in incomplete markets. We also study the
differentiability of the
solutions to these problems with respect to their initial values. We
show that
the key conditions for the results to hold true are that the relative
risk
aversion coefficient of the utility function is uniformly bounded
away from
zero and infinity, and that the prices of traded securities are sigma-
bounded
under the num\'{e}raire given by the optimal wealth process.
http://front.math.ucdavis.edu/math.PR/0610224
---------------------------------------------------------------
5055. A SPATIALLY EXPLICIT MODEL FOR COMPETITION AMONG SPECIALISTS
AND GENERALISTS IN A HETEROGENEOUS ENVIRONMENT
N. Lanchier and C. Neuhauser
Competition is a major force in structuring ecological communities. The
strength of competition can be measured using the concept of a niche.
A niche
comprises the set of requirements of an organism in terms of habitat,
environment and functional role. The more niches overlap, the stronger
competition is. The niche breadth is a measure of specialization: the
smaller
the niche space of an organism, the more specialized the organism is. It
follows that, everything else being equal, generalists tend to be more
competitive than specialists. In this paper, we compare the outcome of
competition among generalists and specialists in a spatial versus a
nonspatial
habitat in a heterogeneous environment. Generalists can utilize the
entire
habitat, whereas specialists are restricted to their preferred
habitat type. We
find that although competitiveness decreases with specialization,
specialists
are more competitive in a spatial than in a nonspatial habitat as
patchiness
increases.
http://front.math.ucdavis.edu/math.PR/0610227
---------------------------------------------------------------
5056. POSITIVE RECURRENCE OF PROCESSES ASSOCIATED TO CRYSTAL GROWTH
MODELS
E. D. Andjel and M. V. Menshikov and V. V. Sisko
We show that certain Markov jump processes associated to crystal growth
models are positive recurrent when the parameters satisfy a rather
natural
condition.
http://front.math.ucdavis.edu/math.PR/0610172
---------------------------------------------------------------
5057. STABILITY AND GENERICITY FOR SPDES DRIVEN BY SPATIALLY
CORRELATED NOISE
K. Bahlali and M. Eddahbi and M. Mellouk
We consider stochastic partial differential equations on $\mathbb{R}^
{d},
d\geq 1$, driven by a Gaussian noise white in time and colored in
space, for
which the pathwise uniqueness holds. By using the Skorokhod
representation
theorem we establish various strong stability results. Then, we give an
application to the convergence of the Picard successive
approximation. Finally,
we show that in the sense of Baire category, almost all stochastic
partial
differential equations with continuous and bounded coefficients have the
properties of existence and uniqueness of solutions as well as the
continuous
dependence on the coefficients.
http://front.math.ucdavis.edu/math.PR/0610174
---------------------------------------------------------------
5058. A DUALITY APPROACH FOR THE WEAK APPROXIMATION OF STOCHASTIC
DIFFERENTIAL EQUATIONS
Emmanuelle Cl\'{e}ment and Arturo Kohatsu-Higa and Damien Lamberton
In this article we develop a new methodology to prove weak approximation
results for general stochastic differential equations. Instead of
using a
partial differential equation approach as is usually done for
diffusions, the
approach considered here uses the properties of the linear equation
satisfied
by the error process. This methodology seems to apply to a large
class of
processes and we present as an example the weak approximation of
stochastic
delay equations.
http://front.math.ucdavis.edu/math.PR/0610178
---------------------------------------------------------------
5059. A NEW COEXISTENCE RESULT FOR COMPETING CONTACT PROCESSES
Benjamin Chan and Richard Durrett
Neuhauser [Probab. Theory Related Fields 91 (1992) 467--506]
considered the
two-type contact process and showed that on $\mathbb{Z}^2$
coexistence is not
possible if the death rates are equal and the particles use the same
dispersal
neighborhood. Here, we show that it is possible for a species with a
long-, but
finite, range dispersal kernel to coexist with a superior competitor
with
nearest-neighbor dispersal in a model that includes deaths of blocks
due to
``forest fires.''
http://front.math.ucdavis.edu/math.PR/0610179
---------------------------------------------------------------
5060. MULTITYPE RANDOMIZED REED--FROST EPIDEMICS AND EPIDEMICS UPON
RANDOM GRAPHS
Peter Neal
We consider a multitype epidemic model which is a natural extension
of the
randomized Reed--Frost epidemic model. The main result is the
derivation of an
asymptotic Gaussian limit theorem for the final size of the epidemic.
The
method of proof is simpler, and more direct, than is used for similar
results
elsewhere in the epidemics literature. In particular, the results are
specialized to epidemics upon extensions of the Bernoulli random graph.
http://front.math.ucdavis.edu/math.PR/0610180
---------------------------------------------------------------
5061. PARALLEL AND INTERACTING MARKOV CHAINS MONTE CARLO METHOD
Fabien Campillo (IRISA / INRIA Rennes) and Vivien Rossi (IURC)
In many situations it is important to be able to propose $N$ independent
realizations of a given distribution law. We propose a strategy for
making $N$
parallel Monte Carlo Markov Chains (MCMC) interact in order to get an
approximation of an independent $N$-sample of a given target law. In
this
method each individual chain proposes candidates for all other
chains. We prove
that the set of interacting chains is itself a MCMC method for the
product of
$N$ target measures. Compared to independent parallel chains this
method is
more time consuming, but we show through concrete examples that it
possesses
many advantages: it can speed up convergence toward the target law as
well as
handle the multi-modal case.
http://front.math.ucdavis.edu/math.PR/0610181
---------------------------------------------------------------
5062. ADAPTIVE POISSON DISORDER PROBLEM
Erhan Bayraktar and Savas Dayanik and Ioannis Karatzas
We study the quickest detection problem of a sudden change in the
arrival
rate of a Poisson process from a known value to an unknown and
unobservable
value at an unknown and unobservable disorder time. Our objective is
to design
an alarm time which is adapted to the history of the arrival process and
detects the disorder time as soon as possible. In previous solvable
versions of
the Poisson disorder problem, the arrival rate after the disorder has
been
assumed a known constant. In reality, however, we may at most have
some prior
information about the likely values of the new arrival rate before
the disorder
actually happens, and insufficient estimates of the new rate after
the disorder
happens. Consequently, we assume in this paper that the new arrival
rate after
the disorder is a random variable. The detection problem is shown to
admit a
finite-dimensional Markovian sufficient statistic, if the new rate has a
discrete distribution with finitely many atoms. Furthermore, the
detection
problem is cast as a discounted optimal stopping problem with running
cost for
a finite-dimensional piecewise-deterministic Markov process. This
optimal
stopping problem is studied in detail in the special case where the
new arrival
rate has Bernoulli distribution. This is a nontrivial optimal
stopping problem
for a two-dimensional piecewise-deterministic Markov process driven
by the same
point process. Using a suitable single-jump operator, we solve it fully,
describe the analytic properties of the value function and the
stopping region,
and present methods for their numerical calculation. We provide a
concrete
example where the value function does not satisfy the smooth-fit
principle on a
proper subset of the connected, continuously differentiable optimal
stopping
boundary, whereas it does on the complement of this set.
http://front.math.ucdavis.edu/math.PR/0610184
---------------------------------------------------------------
5063. LOCAL ALIGNMENT OF MARKOV CHAINS
Niels Richard Hansen
We consider local alignments without gaps of two independent Markov
chains
from a finite alphabet, and we derive sufficient conditions for the
number of
essentially different local alignments with a score exceeding a high
threshold
to be asymptotically Poisson distributed. From the Poisson
approximation a
Gumbel approximation of the maximal local alignment score is
obtained. The
results extend those obtained by Dembo, Karlin and Zeitouni [Ann.
Probab. 22
(1994) 2022--2039] for independent sequences of i.i.d. variables.
http://front.math.ucdavis.edu/math.PR/0610187
---------------------------------------------------------------
5064. COUPLING WITH THE STATIONARY DISTRIBUTION AND IMPROVED SAMPLING
FOR COLORINGS AND INDEPENDENT SETS
Thomas P. Hayes and Eric Vigoda
We present an improved coupling technique for analyzing the mixing
time of
Markov chains. Using our technique, we simplify and extend previous
results for
sampling colorings and independent sets. Our approach uses properties
of the
stationary distribution to avoid worst-case configurations which
arise in the
traditional approach. As an application, we show that for $k/\Delta
>1.764$,
the Glauber dynamics on $k$-colorings of a graph on $n$ vertices with
maximum
degree $\Delta$ converges in $O(n\log n)$ steps, assuming $\Delta =
\Omega(\log
n)$ and that the graph is triangle-free. Previously, girth $\ge 5$
was needed.
As a second application, we give a polynomial-time algorithm for
sampling
weighted independent sets from the Gibbs distribution of the hard-
core lattice
gas model at fugacity $\lambda <(1-\epsilon)e/\Delta$, on a regular
graph $G$
on $n$ vertices of degree $\Delta =\Omega(\log n)$ and girth $\ge 6$.
The best
known algorithm for general graphs currently assumes $\lambda <2/
(\Delta -2)$.
http://front.math.ucdavis.edu/math.PR/0610188
---------------------------------------------------------------
5065. CENTRAL LIMIT THEOREMS FOR GAUSSIAN POLYTOPES
I. Barany and V. H. Vu
Choose $n$ random, independent points in $\R^d$ according to the
standard
normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random
polytope}. We prove that the volume and the number of faces of $K_n$
satisfy
the central limit theorem, settling a well known conjecture in the
field.
http://front.math.ucdavis.edu/math.CO/0610192
---------------------------------------------------------------
5066. $L^P$ PROPERTIES FOR GAUSSIAN RANDOM SERIES
A. Ayache and N. Tzvetkov
We study L^p properties of Gaussian random series with particular
attention
to the case of radial functions.
http://front.math.ucdavis.edu/math.PR/0610139
---------------------------------------------------------------
5067. LIMIT CORRELATION FUNCTIONS FOR FIXED TRACE RANDOM MATRIX
ENSEMBLES
Friedrich G\"otze and Mikhail Gordin
Universal limits for the eigenvalue correlation functions in the bulk
of the
spectrum are shown for a class of nondeterminantal random matrices
known as the
fixed trace ensemble.
http://front.math.ucdavis.edu/math.PR/0610149
---------------------------------------------------------------
5068. POINTS ON HEMISPHERES
Jan Fricke
We will show that for any $n\ge N$ points on the $N$-dimensional
sphere $S^N$
there is a closed hemisphere which contains at least
$\lfloor\frac{n+N+1}{2}\rfloor$ of these points. This bound is sharp
and we
will calculate the amount of sets which realize this value.
If we change to open hemispheres things will be easier. For any $n
$ points on
the sphere there is an open hemisphere which contains at least
$\lfloor\frac{n+1}{2}\rfloor$ of these points, independent of the
dimension.
This bound is sharp.
http://front.math.ucdavis.edu/math.MG/0610140
---------------------------------------------------------------
5069. A MUTATION-SELECTION MODEL FOR GENERAL GENOTYPES WITH
RECOMBINATION
Steven N. Evans and David Steinsaltz and Kenneth W. Wachter
A probability model is presented for the dynamics of mutation-selection
balance in a infinite-population infinite-sites setting sufficiently
general to
cover mutation-driven changes in full age-specific demographic
schedules. An
earlier work by the same authors presented a haploid model -- without
genetic
recombination -- of similar scope. This work complements that model,
adding
genetic recombination, based on a well-known general discrete-population
genetic model of N. Barton and M. Turelli. The model with
recombination is a
flow on Poisson intensities, substantially different from the haploid
model. It
is shown that the new model arises from the haploid model when
recombination is
added, in the limit as generations per unit time go to infinity, and
selection
strength and mutation per generation go to 0.
http://front.math.ucdavis.edu/q-bio.PE/0609046
---------------------------------------------------------------
5070. FLUCTUATION THEORY OF CONNECTIVITIES FOR SUBCRITICAL RANDOM
CLUSTER MODELS
Massimo Campanino and Dmitry Ioffe and Yvan Velenik
We develop a fluctuation theory of connectivities for subcritical random
cluster models. The theory is based on a non-perturbative description
of long
connected clusters in terms of essentially one-dimensional chains of
irreducible objects. Our construction leads to an effective random walk
representation of percolation clusters. The results include a
derivation of a
sharp Ornstein-Zernike type asymptotic formula for 2-point functions,
a proof
of analyticity and strict convexity of inverse correlation length and
a proof
of an invariance principle for connected clusters under diffusive
scaling. In
two dimensions, duality considerations enable a reformulation of
these results
for supercritical nearest-neighbour random cluster measures, in
particular for
nearest-neighbour Potts models in the phase transition regime.
Accordingly, we
prove that equilibrium crystal shapes are always analytic and
strictly convex
and that the interfaces between different phases are always
diffusive. Thus, no
roughening transition is possible in the whole regime where our
results apply.
Our results hold under an assumption of exponential decay of
finite volume
wired connectivities in rectangular domains that is conjectured to
hold in the
whole subcritical regime; the latter is known to be true, in any
dimensions,
when q=1, q=2, and when q is sufficiently large. In two dimensions the
assumption holds whenever there is an exponential decay of
connectivities in
the infinite volume measure. By duality this includes all supercritical
nearest-neighbour Potts models with positive surface tension between
ordered
phases.
http://front.math.ucdavis.edu/math.PR/0610100
---------------------------------------------------------------
5071. REFLECTED DIFFUSIONS DEFINED VIA THE EXTENDED SKOROKHOD MAP
K.Ramanan
This work introduces the extended Skorokhod problem (ESP) and associated
extended Skorokhod map (ESM) that enable a pathwise construction of
reflected
diffusions that are not necessarily semimartingales. Roughly
speaking, given
the closure G of an open connected set in R^J, a non-empty convex
cone d(x) in
R^J, specified at each point x on the boundary of G, and a cadlag
trajectory
\psi taking values in R^J, the ESM defines a constrained version \phi
of \psi
that takes values in G and is such that the increments of \phi - \psi
on any
interval [s,t] lie in the closed convex hull of the directions d(\phi
(u)), u in
(s,t]. General deterministic properties of the ESP are first
established under
the only assumption that the graph of d(.) is closed. Next, for a
class of
multi-dimensional ESPs on polyhedral domains, pathwise uniqueness and
existence
of strong solutions to the associated stochastic differential
equations is
established. In addition, it is also proved that these reflected
diffusions are
semimartingales on [0,\tau_0], where \tau_0 is the time to hit the
set of
points x on the boundary for which d(x) contains a line. One
motivation for the
study of this class of reflected diffusions is that they arise as
approximations of queueing networks in heavy traffic that use the so-
called
generalised processor sharing discipline.
http://front.math.ucdavis.edu/math.PR/0610103
---------------------------------------------------------------
5072. KINETICALLY CONSTRAINED SPIN MODELS
Nicoletta Cancrini and Fabio Martinelli and Cyril Roberto (LAMA)
and Cristina Toninelli (PMA)
We analyze the density and size dependence of the relaxation time for
kinetically constrained spin models (KCSM) intensively studied in the
physical
literature as simple models sharing some of the features of a glass
transition.
KCSM are interacting particle systems on $\Z^d$ with Glauber-like
dynamics,
reversible w.r.t. a simple product i.i.d Bernoulli($p$) measure. The
essential
feature of a KCSM is that the creation/destruction of a particle at a
given
site can occur only if the current configuration of empty sites
around it
satisfies certain constraints which completely define each specific
model. No
other interaction is present in the model. From the mathematical
point of view,
the basic issues concerning positivity of the spectral gap inside the
ergodicity region and its scaling with the particle density $p$
remained open
for most KCSM (with the notably exception of the East model in $d=1$
\cite{Aldous-Diaconis}). Here for the first time we: i) identify the
ergodicity
region by establishing a connection with an associated bootstrap
percolation
model; ii) develop a novel multi-scale approach which proves
positivity of the
spectral gap in the whole ergodic region; iii) establish, sometimes
optimal,
bounds on the behavior of the spectral gap near the boundary of the
ergodicity
region and iv) establish pure exponential decay for the persistence
function.
Our techniques are flexible enough to allow a variety of constraints
and our
findings disprove certain conjectures which appeared in the physical
literature
on the basis of numerical simulations.
http://front.math.ucdavis.edu/math.PR/0610106
---------------------------------------------------------------
5073. ON QUASI SUCCESSFUL COUPLINGS OF MARKOV PROCESSES
Michael Blank and Sergey Pirogov
The notion of a successful coupling of Markov processes, based on the
idea
that both components of the coupled system ``intersect'' in finite
time with
probability one, is extended to cover situations when the coupling is
unnecessarily Markovian and its components are only converging (in a
certain
sense) to each other with time. Under these assumptions the unique
ergodicity
of the original Markov process is proven. A price for this
generalization is
the weak convergence to the unique invariant measure instead of the
strong one.
Applying these ideas to infinite interacting particle systems we
consider even
more involved situations when the unique ergodicity can be proven
only for a
restriction of the original system to a certain class of initial
distributions
(e.g. translational invariant ones). Questions about the existence of
invariant
measures with a given particle density are discussed as well.
http://front.math.ucdavis.edu/math.PR/0610118
---------------------------------------------------------------
5074. ON THE ZERO MASS LIMIT OF TAGGED PARTICLE DIFFUSION IN THE 1-D
RAYLEIGH-GAS
Peter Balint (1) and Balint Toth (1) and Peter Toth (2) ((1)
Institute of Mathematics, Technical University of Budapest, (2)
Renyi Institute, Hungarian
Academy of Sciences)
We consider the M -> 0 limit for tagged particle diffusion in a 1-
dimensional
Rayleigh-gas, studied originaly by Sinai and Soloveichik (1986),
respectively
by Szasz and Toth (1986). In this limit we derive a new type of model
for
tagged paricle diffusion, with Calogero-Moser-Sutherland (i.e. inverse
quadratic) interaction potential between the two central particles.
Computer
simulations on this new model reproduce exactly the numerical value
of the
limiting variance obtained by Boldrighini, Frigio and Tognetti (2002).
http://front.math.ucdavis.edu/math.PR/0610125
---------------------------------------------------------------
5075. BETTER BELL INEQUALITIES (PASSION AT A DISTANCE)
Richard D. Gill
I explain quantum nonlocality experiments and discuss how to optimize
them.
Statistical tools from missing data maximum likelihood are crucial.
New results
are given on CGLMP, CH and ladder inequalities. Open problems are also
discussed.
http://front.math.ucdavis.edu/math.ST/0610115
---------------------------------------------------------------
5076. MOMENT BOUNDS FOR THE SMOLUCHOWSKI EQUATION AND THEIR CONSEQUENCES
Alan Hammond and Fraydoun Rezakhanlou
We prove uniform bounds on moments X_a = \sum_{m}{m^a f_m(x,t)} of the
Smoluchowski coagulation equations with diffusion, valid in any
dimension. If
the collision propensities \alpha(n,m) of mass n and mass m particles
grow more
slowly than (n+m)(d(n) + d(m)), and the diffusion rate d(\cdot) is
non-increasing and satisfies m^{-b_1} \leq d(m) \leq m^{-b_2} for
some b_1 and
b_2 satisfying 0 \leq b_2 < b_1 < \infty, then any weak solution
satisfies X_a
\in L^{\infty}(\mathbb{R}^d \times [0,T]) \cap L^1(\mathbb{R}^d
\times [0,T])
for every a \in \mathbb{N} and T \in (0,\infty), (provided that
certain moments
of the initial data are finite). As a consequence, we infer that these
conditions are sufficient to ensure uniqueness of a weak solution and
its
conservation of mass.
http://front.math.ucdavis.edu/math.AP/0610090
---------------------------------------------------------------
5077. GEODESICS AND ALMOST GEODESIC CYCLES IN RANDOM REGULAR GRAPHS
Itai Benjamini and Carlos Hoppen and Eran ofek and Pawel Pralat
and Nick Wormald
A geodesic in a graph G is a shortest path between two vertices of G.
For a
specific function e(n) of n, we define an almost geodesic cycle C in
G to be a
cycle in which for every two vertices u and v in C, the distance d_G
(u,v) is at
least d_C(u,v)-e(n). Let f(n) be any function tending to infinity
with n. We
consider a random d-regular graph on n vertices. We show that almost
all pairs
of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1}
\log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we
obtain
results on near-geodesic paths. We also give the limiting
distribution of the
number of geodesics between two random vertices in this random graph.
http://front.math.ucdavis.edu/math.MG/0610089
---------------------------------------------------------------
5078. THE EFFECT OF DISORDER ON POLYMER DEPINNING TRANSITIONS
Kenneth S. Alexander
We consider a polymer, with monomer locations modeled by the
trajectory of a
Markov chain, in the presence of a potential that interacts with the
polymer
when it visits a particular site 0. We assume that probability of an
excursion
of length $n$ is given by $n^{-c}\phi(n)$ for some $1<c<2$ and slowly
varying
$\phi$. Disorder is introduced by having the interaction vary from
one monomer
to another, as a constant $u$ plus i.i.d. mean-0 randomness. There is a
critical value of $u$ above which the polymer is pinned, placing a
positive
fraction (called the contact fraction) of its monomers at 0 with high
probability. To see the effect of disorder on the depinning
transition, we
compare the contact fraction and free energy (as functions of $u$) to
the
corresponding annealed system. We show that for $c>3/2$, at high
temperature,
the quenched and annealed curves differ significantly only in a very
small
neighborhood of the critical point--the size of this neighborhood
scales as
$\beta^{1/(2c-3)}$ where $\beta$ is the inverse temperature. For
$c<3/2$, given
$\epsilon>0$, for sufficiently high temperature the quenched and
annealed
curves are within a factor of $1-\epsilon$ for all $u$ near the
critical point;
in particular the quenched and annealed critical points are equal.
For $c=3/2$
the regime depends on the slowly varying function $\phi$.
http://front.math.ucdavis.edu/math.PR/0610008
---------------------------------------------------------------
5079. SOME RELATIONS BETWEEN MUTUAL INFORMATION AND ESTIMATION ERROR
ON WIENER SPACE
Eddy Mayer-Wolf and Moshe Zakai
The model considered is that of "signal plus white noise". Known
connections
between the non-causal filtering error and mutual information are
combined with
new ones involving the causal estimation error, in a general abstract
setup.
The results are shown to be invariant under a wide class of causality
patterns;
they are applied to the derivation of the causal estimation error of
a Gaussian
non-stationary filtering problem and to a multidimensional extension
of the
Yovits-Jackson formula.
http://front.math.ucdavis.edu/math.PR/0610024
---------------------------------------------------------------
5080. QUASIPOTENTIAL AND LOGARITHMIC ASYMPTOTICS OF THE GREEN'S MEASURES
Irina Ignatiouk-Robert
It is proved that the weak large deviation principle of the scaled
processes
$Z^\eps(t) = \eps Z(t/\eps)$ implies the weak large deviation
principle for the
scaled Green's measures of the Markov process $Z(t)$.
http://front.math.ucdavis.edu/math.PR/0610040
---------------------------------------------------------------
5081. A NOTE ON RECURRENT RANDOM WALKS
Dimitrios Cheliotis
For any recurrent random walk (S_n)_{n>0} on R, there are increasing
sequences (g_n)_{n>0} converging to infinity for which (g_n S_n)_
{n>0} has at
least one finite accumulation point. For one class of random walks,
we give a
criterion on (g_n)_{n>0} and the distribution of S_1 determining the
set of
accumulation points for (g_n S_n)_{n>0}. This extends, with a simpler
proof, a
result of K.L. Chung and P. Erdos. Finally, for recurrent, symmetric
random
walks, we give a criterion characterizing the increasing sequences
(g_n)_{n>0}
of positive numbers for which liminf g_n|S_n|=0.
http://front.math.ucdavis.edu/math.PR/0610056
---------------------------------------------------------------
5082. ONE DIMENSIONAL DIFFUSION IN AN ASYMMETRIC RANDOM ENVIRONMENT
Dimitrios Cheliotis
According to a theorem of S. Schumacher, for a diffusion X in an
environment
determined by a stable process that belongs to an appropriate class
and has
index a, it holds that X_t/(log t)^a converges in distribution, as t
goes to
infinity, to a random variable having an explicit description in
terms of the
environment. We compute the density of this random variable in the
case the
stable process is spectrally one-sided. This computation extends a
result of H.
Kesten and quantifies the bias that the asymmetry of the environment
causes to
the behavior of the diffusion.
http://front.math.ucdavis.edu/math.PR/0610057
---------------------------------------------------------------
5083. BERRY-ESSEEN FOR FREE RANDOM VARIABLES
Vladislav Kargin
An analogue of the Berry-Esseen inequality is proved for the speed of
convergence of free additive convolutions of bounded probability
measures. The
obtained rate of convergence is of the order n^{-1/2}, the same as in
the
classical case. An example with binomial measures shows that this
estimate
cannot be improved without imposing further restrictions on convolved
measures.
http://front.math.ucdavis.edu/math.PR/0610072
---------------------------------------------------------------
5084. ON SUPERCONVERGENCE OF SUMS OF FREE RANDOM VARIABLES
Vladislav Kargin
This paper derives sufficient conditions for supeconvergence of sums of
bounded free random variables, and provides an estimate on the rate of
superconvergence.
http://front.math.ucdavis.edu/math.PR/0610075
---------------------------------------------------------------
5085. A SURVEY OF RANDOM PROCESSES WITH REINFORCEMENT
Robin Pemantle
The models surveyed include generalized Polya urns, reinforced random
walks,
interacting urn models, and continuous reinforced processes. Emphasis
is on
methods and results, with sketches provided of some proofs.
Applications are
discussed in statistics, biology, economics and a number of other areas.
http://front.math.ucdavis.edu/math.PR/0610076
---------------------------------------------------------------
5086. PARTIALLY REFLECTED BROWNIAN MOTION: A STOCHASTIC APPROACH TO
TRANSPORT PHENOMENA
Denis S. Grebenkov
Transport phenomena are ubiquitous in nature and known to be
important for
various scientific domains. Examples can be found in physics,
electrochemistry,
heterogeneous catalysis, physiology, etc. To obtain new information
about
diffusive or Laplacian transport towards a semi-permeable or resistive
interface, one can study the random trajectories of diffusing particles
modeled, in a first approximation, by the partially reflected
Brownian motion.
This stochastic process turns out to be a convenient mathematical
foundation
for discrete, semi-continuous and continuous theoretical descriptions of
diffusive transport.
This paper presents an overview of these topics with a special
emphasis on
the close relation between stochastic processes with partial
reflections and
Laplacian transport phenomena. We give selected examples of these
phenomena
followed by a brief introduction to the partially reflected Brownian
motion and
related probabilistic topics (e.g., local time process and spread
harmonic
measure). A particular attention is paid to the use of the Dirichlet-
to-Neumann
operator. Some practical consequences and further perspectives are
discussed.
http://front.math.ucdavis.edu/math.PR/0610080
---------------------------------------------------------------
5087. ROUGH SOLUTIONS FOR THE PERIODIC KORTEWEG-DE VRIES EQUATION
M. Gubinelli
The one dimensional Korteweg-de Vries equation on a periodic domain
and with
initial condition in negative Sobolev spaces is studied using ideas
from the
theory of rough paths. We discuss convergence of Galerkin
approximations and
the presence of a random force of white-noise type in time.
http://front.math.ucdavis.edu/math.AP/0610006
---------------------------------------------------------------
5088. THE PRINCIPLE OF THE LARGE SIEVE
Emmanuel Kowalski
We describe a very general abstract form of sieve based on a large-sieve
inequality which generalizes both the classical sieve inequality of
Montgomery
(and its higher-dimensional variants), and our recent sieve for
Frobenius over
function fields. The general framework suggests new applications. We
get some
first results on the number of prime divisors of ``most'' elements of an
elliptic divisibility sequence, and we develop in some detail
``probabilistic''
sieves for random walks on arithmetic groups, e.g., estimating the
probability
of finding a reducible characteristic polynomial at some step of a
random walk
on SL(n,Z). In addition to the sieve principle, the applications
depend on
bounds for a large sieve constant. To prove such bounds involves a
variety of
deep results, including Property (T) or expanding properties of
Cayley graphs,
and the Riemann Hypothesis over finite fields. It seems likely that
this sieve
can have further applications.
http://front.math.ucdavis.edu/math.NT/0610021
---------------------------------------------------------------
5089. PROBABILITY OF HITTING A DISTANT POINT FOR THE VOTER MODEL
STARTED WITH A SINGLE ONE
Mathieu Merle
The goal of this work is to find the asymptotics of the hitting
probability
of a distant point for the voter model on the integer lattice started
from a
single 1 at the origin. In dimensions 2 or 3, we obtain the precise
asymptotic
behavior of this probability. We use the scaling limit of the voter
model
started from a single 1 at the origin in terms of super-Brownian
motion under
its excursion measure. This invariance principle was stated by
Bramson, Cox and
Le Gall, as a consequence of a theorem of Cox, Durrett and Perkins. Less
precise estimates are derived in dimensions greater than 4.
http://front.math.ucdavis.edu/math.PR/0609826
---------------------------------------------------------------
5090. CONCENTRATION INEQUALITIES FOR DEPENDENT RANDOM VARIABLES VIA
THE MARTINGALE METHOD
Leonid Kontorovich and Kavita Ramanan
We use the martingale method to establish concentration inequalities
for a
class of dependent random sequences on a countable state space, with the
constants in the inequalities expressed in terms of certain mixing
coefficients. Along the way, we obtain bounds on certain martingale
differences
associated with the random sequences, which may be of independent
interest. As
an application of our result, we also derive a concentration
inequality for
inhomogeneous Markov chains, and establish an extremal property
associated with
their martingale difference bounds. This work complements certain
concentration
inequalities obtained by Marton and Samson, while also providing a
different
proof of some known results.
http://front.math.ucdavis.edu/math.PR/0609835
---------------------------------------------------------------
5091. NON-LOCAL DIRICHLET FORMS AND SYMMETRIC JUMP PROCESSES
M.T. Barlow and R.F. Bass and Z.-Q. Chen. and M. Kassmann
We consider the symmetric non-local Dirichlet form $(E, F)$ given by \
[ E
(f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$
the closure
of the set of $C^1$ functions on $R^d$ with compact support with
respect to
$E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where
the jump
kernel $J$ satisfies \[ \kappa_1|y-x|^{-d-\alpha} \leq J(x,y) \leq
\kappa_2|y-x|^{-d-\beta} \] for $0<\alpha< \beta <2, |x-y|<1$. This
assumption
allows the corresponding jump process to have jump intensities whose
size
depends on the position of the process and the direction of the jump.
We prove
upper and lower estimates on the heat kernel. We construct a strong
Markov
process corresponding to $(E, F)$. We prove a parabolic Harnack
inequality for
nonnegative functions that solve the heat equation with respect to $E
$. Finally
we construct an example where the corresponding harmonic functions
need not be
continuous.
http://front.math.ucdavis.edu/math.PR/0609842
---------------------------------------------------------------
5092. RANDOM PATTERNS GENERATED BY RANDOM PERMUTATIONS OF NATURAL
NUMBERS
G.Oshanin (1 and 2) and R.Voituriez (1) and S.Nechaev (3) and
O.Vasilyev (2) and F.Hivert (4)((1) LPTMC, Universite Paris 6,
France; (2) Inhomogeneous
Condensed Matter Department, Max-Planck-Institute Stuttgart,
Germany; (3)
LPTMS, Universite Paris-Sud, France; (4) LITIS/LIFAR, Universite
de Rouen,
France)
We survey recent results on some one- and two-dimensional patterns
generated
by random permutations of natural numbers. In the first part, we discuss
properties of random walks, evolving on a one-dimensional regular
lattice in
discrete time $n$, whose moves to the right or to the left are
induced by the
rise-and-descent sequence associated with a given random permutation. We
determine exactly the probability of finding the trajectory of such a
permutation-generated random walk at site $X$ at time $n$, obtain the
probability measure of different excursions and define the asymptotic
distribution of the number of "U-turns" of the trajectories -
permutation
"peaks" and "through". In the second part, we focus on some statistical
properties of surfaces obtained by randomly placing natural numbers
$1,2,3,
>...,L$ on sites of a 1d or 2d square lattices containing $L$ sites. We
calculate the distribution function of the number of local "peaks" -
sites the
number at which is larger than the numbers appearing at nearest-
neighboring
sites - and discuss some surprising collective behavior emerging in
this model.
http://front.math.ucdavis.edu/cond-mat/0609718
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