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Probability Abstracts 95
This document contains abstracts 4722-5092 from
Oct-1-2006 to Dic-31-2006.
They have been mailed on Jan 1st, 2007.
Author(s): Pablo A. Ferrari and Claudio Landim and Valentin V. Sisko
Abstract: 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://arXiv.org/abs/math/0612856
http://front.math.ucdavis.edu/math.PR/0612856
(alternate)
Author(s): Alan J. Bray and Richard Smith
Abstract: 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://arXiv.org/abs/cond-mat/0612563
http://front.math.ucdavis.edu/cond-mat/0612563
(alternate)
Author(s): Emmanuel J. Candes and Paige A. Randall
Abstract: 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://arXiv.org/abs/cs/0612124
http://front.math.ucdavis.edu/cs.IT/0612124
(alternate)
Author(s): Francis Comets and Serguei Popov and Gunter Sch\"utz and Marina Vachkovskaia
Abstract: 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://arXiv.org/abs/math/0612799
http://front.math.ucdavis.edu/math.PR/0612799
(alternate)
Author(s): Olle H\"{a}ggstr\"{o}m and Johan Jonasson
Abstract: 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://arXiv.org/abs/math/0612812
http://front.math.ucdavis.edu/math.PR/0612812
(alternate)
Author(s): Svante Janson and Malwina J. Luczak
Abstract: 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://arXiv.org/abs/math/0612827
http://front.math.ucdavis.edu/math.PR/0612827
(alternate)
Author(s): N. K. Jana and B. V. Rao
Abstract: 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://arXiv.org/abs/math/0612836
http://front.math.ucdavis.edu/math.PR/0612836
(alternate)
Author(s): Evarist Gin\'{e} and Vladimir Koltchinskii and Wenbo Li and Joel Zinn
Abstract: 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://arXiv.org/abs/math/0612726
http://front.math.ucdavis.edu/math.PR/0612726
(alternate)
Author(s): Ou Zhao and Michael Woodroofe
Abstract: 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://arXiv.org/abs/math/0612747
http://front.math.ucdavis.edu/math.PR/0612747
(alternate)
Author(s): Nathanael Enriquez (PMA)
Abstract: 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://arXiv.org/abs/math/0612770
http://front.math.ucdavis.edu/math.PR/0612770
(alternate)
Author(s): Evarist Gin\'{e} and Vladimir Koltchinskii
Abstract: 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://arXiv.org/abs/math/0612777
http://front.math.ucdavis.edu/math.PR/0612777
(alternate)
Author(s): stefano m. iacus and nakahiro yoshida
Abstract: 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://arXiv.org/abs/math/0612784
http://front.math.ucdavis.edu/math.PR/0612784
(alternate)
Author(s): Greg W Anderson and Ofer Zeitouni
Abstract: 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://arXiv.org/abs/math/0612791
http://front.math.ucdavis.edu/math.PR/0612791
(alternate)
Author(s): Janos Englander
Abstract: 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://arXiv.org/abs/math/0612797
http://front.math.ucdavis.edu/math.PR/0612797
(alternate)
Author(s): Zhenxin Liu
Abstract: 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://arXiv.org/abs/math/0612774
http://front.math.ucdavis.edu/math.DS/0612774
(alternate)
Author(s): E. Ben-Naim and N.W. Hengartner
Abstract: 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://arXiv.org/abs/physics/0612217
http://front.math.ucdavis.edu/physics/0612217
(alternate)
Author(s): Paavo Salminen and Pierre Vallois and Marc Yor
Abstract: 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://arXiv.org/abs/math/0612687
http://front.math.ucdavis.edu/math.PR/0612687
(alternate)
Author(s): Przemys{\l}aw Repetowicz and Peter Richmond
Abstract: 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://arXiv.org/abs/math/0612691
http://front.math.ucdavis.edu/math.PR/0612691
(alternate)
Author(s): Wei Biao Wu
Abstract: 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://arXiv.org/abs/math/0612692
http://front.math.ucdavis.edu/math.PR/0612692
(alternate)
Author(s): Paul Deheuvels
Abstract: 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://arXiv.org/abs/math/0612693
http://front.math.ucdavis.edu/math.PR/0612693
(alternate)
Author(s): Vladimir Dobri\'{c} and Francisco M. Ojeda
Abstract: 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://arXiv.org/abs/math/0612694
http://front.math.ucdavis.edu/math.PR/0612694
(alternate)
Author(s): Raouf Ghomrasni
Abstract: We show that for a wide class of functions $F$ that: $$ {\lim_{\epsilon
\downarrow 0} {\frac{1}{\epsilon}} \int_0^t \Big\{F(s, X_s) - F(s, X_s -
\epsilon)\Big\} d\big_s} = - \int_0^t\int_{\R} F(s, x) d L_s^x $$
where $X_t$ is a continuous semi-martingale, $(L_t^x, x \in \R, t \geq 0)$ its
local time process and $(\big_t, t \geq 0)$ its quadratic variation
process.
http://arXiv.org/abs/math/0612699
http://front.math.ucdavis.edu/math.PR/0612699
(alternate)
Author(s): Dongsheng Wu and Yimin Xiao
Abstract: 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://arXiv.org/abs/math/0612700
http://front.math.ucdavis.edu/math.PR/0612700
(alternate)
Author(s): Shahar Mendelson and Joel Zinn
Abstract: 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://arXiv.org/abs/math/0612703
http://front.math.ucdavis.edu/math.PR/0612703
(alternate)
Author(s): Richard Nickl
Abstract: 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://arXiv.org/abs/math/0612706
http://front.math.ucdavis.edu/math.PR/0612706
(alternate)
Author(s): Magda Peligrad and Sergey Utev
Abstract: 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://arXiv.org/abs/math/0612707
http://front.math.ucdavis.edu/math.PR/0612707
(alternate)
Author(s): Jean Bertoin (PMA and DMA)
Abstract: 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://arXiv.org/abs/math/0612710
http://front.math.ucdavis.edu/math.PR/0612710
(alternate)
Author(s): Adrian P.C. Lim
Abstract: 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://arXiv.org/abs/math/0612711
http://front.math.ucdavis.edu/math.PR/0612711
(alternate)
Author(s): Jos\'{e} E. Figueroa-L\'{o}pez and Christian Houdr\'{e}
Abstract: 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://arXiv.org/abs/math/0612697
http://front.math.ucdavis.edu/math.ST/0612697
(alternate)
Author(s): Philippe Berthet and David M. Mason
Abstract: 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://arXiv.org/abs/math/0612701
http://front.math.ucdavis.edu/math.ST/0612701
(alternate)
Author(s): Miguel A. Arcones
Abstract: We find the Bahadur slope of the Lilliefors and Cram\'{e}r--von Mises tests
of normality.
http://arXiv.org/abs/math/0612708
http://front.math.ucdavis.edu/math.ST/0612708
(alternate)
Author(s): Kenneth S. Alexander
Abstract: 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://arXiv.org/abs/math/0612625
http://front.math.ucdavis.edu/math.PR/0612625
(alternate)
Author(s): Tatyana S. Turova and Thomas Vallier
Abstract: 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
http://arXiv.org/abs/math/0612644
http://front.math.ucdavis.edu/math.PR/0612644
(alternate)
Author(s): Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: 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://arXiv.org/abs/math/0612648
http://front.math.ucdavis.edu/math.PR/0612648
(alternate)
Author(s): Aur\'{e}lien Alfonsi (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: 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://arXiv.org/abs/math/0612649
http://front.math.ucdavis.edu/math.PR/0612649
(alternate)
Author(s): Marie F. Kratz
Abstract: 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://arXiv.org/abs/math/0612577
http://front.math.ucdavis.edu/math.PR/0612577
(alternate)
Author(s): Charles Bordenave and Serguei Foss and Vsevolod Shneer
Abstract: 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://arXiv.org/abs/math/0612583
http://front.math.ucdavis.edu/math.PR/0612583
(alternate)
Author(s): David Croydon
Abstract: 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://arXiv.org/abs/math/0612585
http://front.math.ucdavis.edu/math.PR/0612585
(alternate)
Author(s): Igor S. Borisov and Alexander A. Bystrov
Abstract: 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://arXiv.org/abs/math/0612594
http://front.math.ucdavis.edu/math.PR/0612594
(alternate)
Author(s): Tom Kennedy
Abstract: 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://arXiv.org/abs/math/0612609
http://front.math.ucdavis.edu/math.PR/0612609
(alternate)
Author(s): Yoshiyasu Ishigami
Abstract: 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://arXiv.org/abs/math/0612601
http://front.math.ucdavis.edu/math.CO/0612601
(alternate)
Author(s): Jonathan E. Taylor and Robert J. Adler
Abstract: 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://arXiv.org/abs/math/0612580
http://front.math.ucdavis.edu/math.DG/0612580
(alternate)
Author(s): Hari Bercovici and Jiun-Chau Wang
Abstract: 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://arXiv.org/abs/math/0612599
http://front.math.ucdavis.edu/math.OA/0612599
(alternate)
Author(s): Dimitrios Cheliotis
Abstract: 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://arXiv.org/abs/math/0612533
http://front.math.ucdavis.edu/math.PR/0612533
(alternate)
Author(s): V. P. Maslov and T. V. Maslova
Abstract: 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://arXiv.org/abs/math/0612540
http://front.math.ucdavis.edu/math.PR/0612540
(alternate)
Author(s): David Gamarnik and Sean Meyn
Abstract: 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://arXiv.org/abs/math/0612544
http://front.math.ucdavis.edu/math.PR/0612544
(alternate)
Author(s): Thomas M. Liggett and Rinaldo B. Schinazi and and Jason Schweinsberg
Abstract: 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://arXiv.org/abs/math/0612564
http://front.math.ucdavis.edu/math.PR/0612564
(alternate)
Author(s): Walid Hachem (LTCI) and Oleksiy Khorunzhiy and Philippe Loubaton (IGM-LabInfo), Jamal Najim (LTCI), Leonid Pastur
Abstract: 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://arXiv.org/abs/cs/0612076
http://front.math.ucdavis.edu/cs.IT/0612076
(alternate)
Author(s): Richard G. Clegg
Abstract: 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://arXiv.org/abs/math/0612476
http://front.math.ucdavis.edu/math.PR/0612476
(alternate)
Author(s): Vladislav Vysotsky
Abstract: 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://arXiv.org/abs/math/0612490
http://front.math.ucdavis.edu/math.PR/0612490
(alternate)
Author(s): Gilles Pag\`{e}s (PMA)
Abstract: 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://arXiv.org/abs/math/0612523
http://front.math.ucdavis.edu/math.PR/0612523
(alternate)
Author(s): Dirk Tasche
Abstract: 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://arXiv.org/abs/math/0612470
http://front.math.ucdavis.edu/math.ST/0612470
(alternate)
Author(s): Bo'az Klartag and Gady Kozma
Abstract: 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://arXiv.org/abs/math/0612517
http://front.math.ucdavis.edu/math.MG/0612517
(alternate)
Author(s): Gerard Ben Arous and Veronique Gayrard and Alexey Kuptsov
Abstract: 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://arXiv.org/abs/math/0612373
http://front.math.ucdavis.edu/math.PR/0612373
(alternate)
Author(s): Maury Bramson and Ofer Zeitouni
Abstract: 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://arXiv.org/abs/math/0612382
http://front.math.ucdavis.edu/math.PR/0612382
(alternate)
Author(s): Mark Adler & Pierre van Moerbeke
Abstract: 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://arXiv.org/abs/math/0612393
http://front.math.ucdavis.edu/math.PR/0612393
(alternate)
Author(s): V. P. Maslov
Abstract: The notion of density of a finite set is discussed. We proof a general
theorem of set theory which refines Bose-Einstein distribution.
http://arXiv.org/abs/math/0612394
http://front.math.ucdavis.edu/math.PR/0612394
(alternate)
Author(s): Hamed Hatami
Abstract: 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://arXiv.org/abs/math/0612405
http://front.math.ucdavis.edu/math.PR/0612405
(alternate)
Author(s): S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA)
Abstract: 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://arXiv.org/abs/math/0612413
http://front.math.ucdavis.edu/math.PR/0612413
(alternate)
Author(s): K.D. Elworthy and Xue-Mei Li
Abstract: 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://arXiv.org/abs/math/0612416
http://front.math.ucdavis.edu/math.PR/0612416
(alternate)
Author(s): Jean-Philippe Anker (MAPMO) and Bruno Schapira (MAPMO and PMA) and Bartosz Trojan (MAPMO)
Abstract: 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://arXiv.org/abs/math/0612385
http://front.math.ucdavis.edu/math.CA/0612385
(alternate)
Author(s): Hamed Hatami and Michael Molloy
Abstract: 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://arXiv.org/abs/math/0612391
http://front.math.ucdavis.edu/math.CO/0612391
(alternate)
Author(s): V. P. Maslov
Abstract: We give a risk-minimizing formula for government investments taking into
account the zero intelligence law for financial markets.
http://arXiv.org/abs/math/0612395
http://front.math.ucdavis.edu/math.GM/0612395
(alternate)
Author(s): Liqun Wang and Klaus P\"otzelberger
Abstract: 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://arXiv.org/abs/math/0612337
http://front.math.ucdavis.edu/math.PR/0612337
(alternate)
Author(s): Jir\^o Akahori and Takahiro Tsuchiya
Abstract: 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://arXiv.org/abs/math/0612341
http://front.math.ucdavis.edu/math.PR/0612341
(alternate)
Author(s): Christian houdr\'e and Trevis J. Litherland
Abstract: 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://arXiv.org/abs/math/0612364
http://front.math.ucdavis.edu/math.PR/0612364
(alternate)
Author(s): Zongxia Liang and Tusheng Zhang
Abstract: 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://arXiv.org/abs/math/0612294
http://front.math.ucdavis.edu/math.PR/0612294
(alternate)
Author(s): Marc Peign\'e and Wolfgang Woess
Abstract: 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://arXiv.org/abs/math/0612306
http://front.math.ucdavis.edu/math.PR/0612306
(alternate)
Author(s): Vincent Bansaye (PMA)
Abstract: 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://arXiv.org/abs/math/0612312
http://front.math.ucdavis.edu/math.PR/0612312
(alternate)
Author(s): J. Creutzig and T. Mueller-Gronbach and K. Ritter
Abstract: 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://arXiv.org/abs/math/0612313
http://front.math.ucdavis.edu/math.PR/0612313
(alternate)
Author(s): J.F. Le Gall and F. Paulin
Abstract: 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://arXiv.org/abs/math/0612315
http://front.math.ucdavis.edu/math.PR/0612315
(alternate)
Author(s): Boris Tsirelson
Abstract: 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://arXiv.org/abs/math/0612303
http://front.math.ucdavis.edu/math.OA/0612303
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: 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://arXiv.org/abs/math/0612224
http://front.math.ucdavis.edu/math.PR/0612224
(alternate)
Author(s): Vladislav Vysotsky
Abstract: 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://arXiv.org/abs/math/0612253
http://front.math.ucdavis.edu/math.PR/0612253
(alternate)
Author(s): Bruno Schapira (MAPMO and PMA)
Abstract: 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://arXiv.org/abs/math/0612272
http://front.math.ucdavis.edu/math.PR/0612272
(alternate)
Author(s): Nicolas Bouleau (CERMICS) and Christophe Chorro (CERMICS and CERMSEM)
Abstract: 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://arXiv.org/abs/math/0612258
http://front.math.ucdavis.edu/math.ST/0612258
(alternate)
Author(s): M. Anoussis and D. Gatzouras
Abstract: 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://arXiv.org/abs/math/0612262
http://front.math.ucdavis.edu/math.FA/0612262
(alternate)
Author(s): Hari Bercovici and Jiun-Chau Wang
Abstract: 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://arXiv.org/abs/math/0612278
http://front.math.ucdavis.edu/math.OA/0612278
(alternate)
Author(s): 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)
Abstract: 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://arXiv.org/abs/cs/0111054
http://front.math.ucdavis.edu/cs.CC/0111054
(alternate)
Author(s): Rudi Cilibrasi and Paul M.B. Vitanyi
Abstract: 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://arXiv.org/abs/cs/0606048
http://front.math.ucdavis.edu/cs.DS/0606048
(alternate)
Author(s): James Allen Fill and David B. Wilson
Abstract: 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://arXiv.org/abs/math/0612205
http://front.math.ucdavis.edu/math.PR/0612205
(alternate)
Author(s): Pierre Andreoletti (MAPMO)
Abstract: 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://arXiv.org/abs/math/0612209
http://front.math.ucdavis.edu/math.PR/0612209
(alternate)
Author(s): Jak\v{s}a Cvitani\'{c} and Robert Liptser and Boris Rozovskii
Abstract: 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://arXiv.org/abs/math/0612212
http://front.math.ucdavis.edu/math.PR/0612212
(alternate)
Author(s): Arvind Singh (PMA)
Abstract: 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://arXiv.org/abs/math/0612220
http://front.math.ucdavis.edu/math.PR/0612220
(alternate)
Author(s): C. Giardin\'a and J. Kurchan and F. Redig
Abstract: 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://arXiv.org/abs/cond-mat/0612198
http://front.math.ucdavis.edu/cond-mat/0612198
(alternate)
Author(s): T. Byczkowski and M. Ryznar and J. Malecki
Abstract: 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://arXiv.org/abs/math/0612176
http://front.math.ucdavis.edu/math.PR/0612176
(alternate)
Author(s): Tamer Oraby
Abstract: 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://arXiv.org/abs/math/0612177
http://front.math.ucdavis.edu/math.PR/0612177
(alternate)
Author(s): Marie-Amelie Morlais
Abstract: 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://arXiv.org/abs/math/0612181
http://front.math.ucdavis.edu/math.PR/0612181
(alternate)
Author(s): C. P. Hughes and A. Nikeghbali and M. Yor
Abstract: 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://arXiv.org/abs/math/0612195
http://front.math.ucdavis.edu/math.PR/0612195
(alternate)
Author(s): Victor Goodman (Indiana University)
Abstract: 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://arXiv.org/abs/math/0612160
http://front.math.ucdavis.edu/math.PR/0612160
(alternate)
Author(s): Michel Fliess (INRIA Futurs)
Abstract: 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://arXiv.org/abs/quant-ph/0612033
http://front.math.ucdavis.edu/quant-ph/0612033
(alternate)
Author(s): Yves Le Jan (LM-Orsay)
Abstract: The purpose of this note is to explore some simple relations between loop
measures, determinants, and Gaussian Markov fields.
http://arXiv.org/abs/math/0612112
http://front.math.ucdavis.edu/math.PR/0612112
(alternate)
Author(s): Anders Johansson and Anders \"Oberg
Abstract: 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://arXiv.org/abs/math/0612131
http://front.math.ucdavis.edu/math.DS/0612131
(alternate)
Author(s): I. Kontoyiannis and S.P. Meyn
Abstract: 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://arXiv.org/abs/math/0612040
http://front.math.ucdavis.edu/math.PR/0612040
(alternate)
Author(s): Elchanan Mossel and Sebastien Roch
Abstract: 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://arXiv.org/abs/math/0612046
http://front.math.ucdavis.edu/math.PR/0612046
(alternate)
Author(s): C. Cotar and V. Limic
Abstract: 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://arXiv.org/abs/math/0612048
http://front.math.ucdavis.edu/math.PR/0612048
(alternate)
Author(s): Andreas Eberle and Carlo Marinelli
Abstract: 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://arXiv.org/abs/math/0612074
http://front.math.ucdavis.edu/math.PR/0612074
(alternate)
Author(s): Dimitris Bertsimas and Natasha Bushueva
Abstract: 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://arXiv.org/abs/math/0612075
http://front.math.ucdavis.edu/math.PR/0612075
(alternate)
Author(s): Konstantin Avrachenkov and Nelly Litvak and Kim Son Pham
Abstract: 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://arXiv.org/abs/math/0612079
http://front.math.ucdavis.edu/math.PR/0612079
(alternate)
Author(s): Christophe Bahadoran
Abstract: 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://arXiv.org/abs/math/0612094
http://front.math.ucdavis.edu/math.PR/0612094
(alternate)
Author(s): Joel A. Tropp
Abstract: 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://arXiv.org/abs/math/0612070
http://front.math.ucdavis.edu/math.MG/0612070
(alternate)
Author(s): Jacques Franchi
Abstract: 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://arXiv.org/abs/math/0612020
http://front.math.ucdavis.edu/math.PR/0612020
(alternate)
Author(s): Franck Barthe (LSProba) and Cyril Roberto (LAMA)
Abstract: 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://arXiv.org/abs/math/0612026
http://front.math.ucdavis.edu/math.PR/0612026
(alternate)
Author(s): Victor Goodman and Kyounghee Kim
Abstract: 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://arXiv.org/abs/math/0612034
http://front.math.ucdavis.edu/math.PR/0612034
(alternate)
Author(s): Victor Goodman and Kyounghee Kim
Abstract: 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://arXiv.org/abs/math/0612035
http://front.math.ucdavis.edu/math.PR/0612035
(alternate)
Author(s): Jesper Lykke Jacobsen (LPTMS and SPhT) and Hubert Saleur (SPhT)
Abstract: 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://arXiv.org/abs/math-ph/0611078
http://front.math.ucdavis.edu/math-ph/0611078
(alternate)
Author(s): R. Husseini and M. Kassmann
Abstract: 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://arXiv.org/abs/math/0611934
http://front.math.ucdavis.edu/math.PR/0611934
(alternate)
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS) and Benjamin Jourdain (CERMICS)
Abstract: 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://arXiv.org/abs/math/0611949
http://front.math.ucdavis.edu/math.PR/0611949
(alternate)
Author(s): Emmanuel Candes and Justin Romberg
Abstract: 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://arXiv.org/abs/math/0611957
http://front.math.ucdavis.edu/math.ST/0611957
(alternate)
Author(s): Sourav Chatterjee and Ron Peled and Yuval Peres and Dan Romik
Abstract: 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://arXiv.org/abs/math/0611886
http://front.math.ucdavis.edu/math.PR/0611886
(alternate)
Author(s): Yulia Mishura and Esko Valkeila
Abstract: We extend the classical Levy characterization of Brownian motion to
fractional Brownian motion.
http://arXiv.org/abs/math/0611913
http://front.math.ucdavis.edu/math.PR/0611913
(alternate)
Author(s): Brice Franke
Abstract: 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://arXiv.org/abs/math/0611852
http://front.math.ucdavis.edu/math.PR/0611852
(alternate)
Author(s): Elise Janvresse (LMRS) and Beno\^{i}t Rittaud (IG) and Thierry De La Rue (LMRS)
Abstract: 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
http://arXiv.org/abs/math/0611860
http://front.math.ucdavis.edu/math.PR/0611860
(alternate)
Author(s): Nizar Demni (PMA)
Abstract: 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://arXiv.org/abs/math/0611863
http://front.math.ucdavis.edu/math.PR/0611863
(alternate)
Author(s): Shige Peng and Mingyu Xu
Abstract: Here we study a new numerical method for BSDE. Then we present a package of
our numerical algorithms of BSDE with convenient user--mac | |
