[PAS] Probability Abstract 96
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Thu Mar 1 08:44:49 CET 2007
Probability Abstracts 96
This document contains abstracts 5093-5304 from
Jan-1-2007 to Feb-28-2007.
They have been mailed on March 1st, 2007.
This letter can be also found on line at
http://lists.imstat.org/PAS/Letters/letter_96.shtml
Please notice that since february 2007, PAS web site and
PAS mailing list have been moved under the mailing list
server of the Institute of Mathematical Statistics
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5093. EXPECTED NUMBER OF SLOPE CROSSINGS OF CERTAIN GAUSSIAN RANDOM
POLYNOMIALS
S. Rezakhah and S. Shemehsavar
Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random polynomial where the
coefficients
$A_0,A_1,... $ form a sequence of centered Gaussian random variables.
Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$,
$j=0,1,2,...$ are
independent, assuming $A_{-1}=0$. The coefficients can be considered
as $n$
consecutive observations of a Brownian motion. We study the number of
times
that such a random polynomial crosses a line which is not necessarily
parallel
to the x-axis. More precisely we obtain the asymptotic behavior of
the expected
number of real roots of the equation $Q_n(x)=Kx$, for the cases that
$K$ is any
non-zero real constant $K=o(n^{1/4})$, and $K=o(n^{1/2})$ separately.
http://front.math.ucdavis.edu/math.PR/0701019
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5094. TILTED STABLE SUBORDINATORS, GAMMA TIME CHANGES AND OCCUPATION
TIME OF RAYS BY BESSEL SPIDERS
Lancelot F. James and Marc Yor
We exhibit, in the form of some identities in law, some connections
between
tilted stable subordinators, time-changed by independent Gamma
processes and
the occupation times of Bessel spiders, or their bridges. These
identities in
law are then explained thanks to excursion theory.
http://front.math.ucdavis.edu/math.PR/0701049
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5095. INTRACTABILITY RATE OF APPROXIMATION PROBLEM FOR RANDOM FIELDS
IN INCREASING DIMENSION
N. Serdyukova
The behavior of average approximation cardinality for d-parametric
random
fields of tensor product type is investigated. The exact rate of
dimension
curse is obtained.
http://front.math.ucdavis.edu/math.PR/0701058
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5096. GOODNESS OF FIT TEST FOR ERGODIC DIFFUSION PROCESSES
Ilia Negri and Yoichi Nishiyama
A goodness of fit test for the drift coefficient of an ergodic diffusion
process is presented. The test is based on the score marked empirical
process.
The weak convergence of the proposed test statistic is studied under
the null
hypotheses and it is proved that the limit process is a continuous
Gaussian
process. The structure of its covariance function allows to calculate
the limit
distribution and it turns out that it is a function of a standard
Brownian
motion and so exact reject regions can be constructed. The proposed
test is
asymptotically distribution free and it is consistent under any
simple fixed
alternative.
http://front.math.ucdavis.edu/math.ST/0701022
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5097. A LAW OF LARGE NUMBERS FOR AN INTERACTING PARTICLE SYSTEM WITH
CONFINING POTENTIAL
Matteo Ortisi (Dept. of Mathematics and University of Milano)
In this paper we consider an interacting particle system modeled as a
system
of $N$ stochastic differential equations driven by Brownian motions
with a
drift term including a confining potential acting on each particle,
and an
interaction potential modeling the interaction among all the
particles of the
system. The limiting behavior as the size $N$ grows to infinity is
achieved as
a law of large numbers for the empirical process associated with the
interacting particle system
http://front.math.ucdavis.edu/math.PR/0701095
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5098. COMPRESSED SENSING AND REDUNDANT DICTIONARIES
Holger Rauhut and Karin Schnass and Pierre Vandergheynst
This article extends the concept of compressed sensing to signals
that are
not sparse in an orthonormal basis but rather in a redundant
dictionary. It is
shown that a matrix, which is a composition of a random matrix of
certain type
and a deterministic dictionary, has small restricted isometry
constants. Thus,
signals that are sparse with respect to the dictionary can be
recovered via
Basis Pursuit from a small number of random measurements. Further,
thresholding
is investigated as recovery algorithm for compressed sensing and
conditions are
provided that guarantee reconstruction with high probability. The
different
schemes are compared by numerical experiments.
http://front.math.ucdavis.edu/math.PR/0701131
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5099. SHORT-LENGTH ROUTES IN LOW-COST NETWORKS VIA POISSON LINE PATTERNS
David J. Aldous and Wilfrid S. Kendall
In designing a network to link n cities in a square of area n, one
might be
guided by the following two desiderata. First, the total network
length should
not be much greater than the length of the shortest network
connecting all
cities. Second, the average route length (taken over source-
destination pairs)
should not be much greater than the average straight-line distance.
How small
can we make these two differences? For typical configurations the
shortest
network length is order n and the average straight-line distance is
order
n^1/2, so it seems implausible that one can construct a network in
which the
first difference is o(n) and the second difference is o(n^1/2). But
in fact one
can do better: for an arbitrary configuration one can construct a
network where
the first difference is o(n) and the second difference is almost as
small as
O(log n). The construction is conceptually simple: over the minimum-
length
connected network (Steiner tree) superimpose a sparse stationary and
isotropic
Poisson line process. The key ingredient is a new result about the
Poisson line
process. Consider two points at distance r apart, and delete from the
line
process all lines which separate these two points. The resulting
pattern of
lines partitions the plane into cells; the cell containing the two
points has
mean boundary length 2r + C log r. Turning to lower bounds we show
that, under
a weak equidistribution assumption, if the first difference is o(n)
then the
second difference cannot be O(sqrt(log n)).
http://front.math.ucdavis.edu/math.PR/0701140
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5100. THE CENTRAL LIMIT THEOREM FOR LS ESTIMATOR IN SIMPLE LINEAR EV
REGRESSION MODELS
Yu Miao and Guangyu Yang and Luming Shen
In this paper, we obtain the central limit theorems for LS estimator in
simple linear errors-in-variables (EV) regression models under some mild
conditions. And we also show that those conditions are necessary in
some sense.
http://front.math.ucdavis.edu/math.PR/0701162
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5101. THE LAW OF THE ITERATED LOGARITHM FOR ADDITIVE FUNCTIONALS OF
MARKOV CHAINS
Yu Miao and Guangyu Yang
In the paper, the law of the iterated logarithm for additive
functionals of
Markov chains is obtained under some weak conditions, which are
weaker than the
conditions of invariance principle of additive functionals of Markov
chains in
M. Maxwell and M. Woodroofe (2000). The main technique is the martingale
argument and the theory of fractional coboundaries.
http://front.math.ucdavis.edu/math.PR/0701167
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5102. ONE DIMENSIONAL NEAREST NEIGHBOR EXCLUSION PROCESSES IN
INHOMOGENEOUS AND RANDOM ENVIRONMENTS
Lincoln Chayes and Thomas M. Liggett
The processes described in the title always have reversible stationary
distributions. In this paper, we give sufficient conditions for the
existence
of, and for the nonexistence of, nonreversible stationary
distributions. In the
case of an i.i.d. environment, these combine to give a necessary and
sufficient
condition for the existence of nonreversible stationary distributions.
http://front.math.ucdavis.edu/math.PR/0701180
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5103. A NOTE FOR EXTENSION OF ALMOST SURE CENTRAL LIMIT THEORY
Yu Miao and Guangyu Yang
H\"ormann (2006) gave an extension of almost sure central limit
theorem for
bounded Lipschitz 1 function. In this paper, we show that his result
of almost
sure central limit theorem is also hold for any Lipschitz function under
stronger conditions.
http://front.math.ucdavis.edu/math.PR/0701183
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5104. ON A TYPE SOBOLEV INEQUALITY AND ITS APPLICATIONS
Witold Bednorz
In the paper we pursue the analysis from the section 5 of the
Talagrand's
paper "Sample boundedness of stochastic processes under increment
conditions."
Ann. Probab. 18, No. 1, 1-49. In particular we give the proof of some
Sobolev
Inequality and then apply it to obtain if and only if condition for all
processes with bounded icrements to have bounded samples. The
processes are
defined on a compact, concave subspaces of $\R^n$ with a metric
$d(s,t)=\eta(||s-t||)$, where $\eta$ is concave and $||.||$ is a norm on
$\R^n$.
http://front.math.ucdavis.edu/math.PR/0701191
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5105. DIFFUSION LIMITED AGGREGATION ON A CYLINDER
Itai Benjamini and Ariel Yadin
We consider the DLA process on a cylinder $G \times \N$. It is shown
that
this process ``grows arms'', provided that the base graph $G$ has
small enough
mixing time. Specifically, if the mixing time of $G$ is at most
$\log^{(2-\eps)}\abs{G}$, the time it takes the cluster to reach the
$m$-th
layer of the cylinder is at most of order $m \cdot \frac{\abs{G}}{\log
\log
\abs{G}}$. In particular we get examples of infinite Cayley graphs of
degree 5,
for which the DLA cluster on these graphs has arbitrarily small density.
In addition, we provide an upper bound on the rate at which the
``arms''
grow. This bound is valid for a large class of base graphs $G$,
including
discrete tori of dimension at least 3.
It is also shown that for any base graph $G$, the density of the
DLA process
on a $G$-cylinder is related to the rate at which the arms of the
cluster grow.
This implies, that for any vertex transitive $G$, the density of DLA
on a
$G$-cylinder is bounded by 2/3.
http://front.math.ucdavis.edu/math.PR/0701201
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5106. ON THE CONSTRUCTIONS OF THE SKEW BROWNIAN MOTION
Antoine Lejay
This article summarizes the various ways one may use to construct the
Skew
Brownian motion, and shows their connections. Recent applications of
this
process in modelling and numerical simulation motivates this survey.
This
article ends with a brief account of related results, extensions and
applications of the Skew Brownian motion.
http://front.math.ucdavis.edu/math.PR/0701219
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5107. A MARKOV CHAIN MODEL OF A POLLING SYSTEM WITH PARAMETER
REGENERATION
Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and and
Serguei Popov
We study a model of a polling system i.e. a collection of $d$ queues
with a
single server that switches from queue to queue. The service time
distribution
and arrival rates change randomly every time a queue is emptied. This
model is
mapped to a mathematically equivalent model of a random walk with
random choice
of transition probabilities, a model which is of independent
interest. All our
results are obtained using methods from the constructive theory of
Markov
chains. We determine conditions for the existence of polynomial
moments of
hitting times for the random walk. An unusual phenomenon of thickness
of the
region of null recurrence for both the random walk and the queueing
model is
also proved.
http://front.math.ucdavis.edu/math.PR/0701226
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5108. FUNCTIONAL CLT FOR RANDOM WALK AMONG BOUNDED RANDOM CONDUCTANCES
Marek Biskup and Timothy M. Prescott
We consider the nearest-neighbor simple random walk on $\Z^d$, $d\ge2$,
driven by a field of i.i.d. random nearest-neighbor conductances
$\omega_{xy}\in[0,1]$. Apart from the requirement that the bonds with
positive
conductances percolate, we pose no restriction on the law of the $
\omega$'s. We
prove that, for a.e. realization of the environment, the path
distribution of
the walk converges weakly to that of non-degenerate, isotropic
Brownian motion.
This holds despite the fact that the local CLT may fail in $d\ge5$
due to
anomalously slow decay of the probability that the walk returns to
the starting
point at a given time (cf math.PR/0611666).
http://front.math.ucdavis.edu/math.PR/0701248
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5109. DIFFUSIVITY IN ONE-DIMENSIONAL GENERALIZED MOTT VARIABLE-RANGE
HOPPING MODELS
Pietro Caputo and Alessandra Faggionato
We consider random walks in random environment which are generalized
versions
of well known effective models for Mott variable--range hopping. We
study the
homogenized diffusion constant of the random walk in the one--
dimensional case.
We prove various estimates on the the low--temperature behavior which
confirm
and extend previous work by physicists.
http://front.math.ucdavis.edu/math.PR/0701253
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5110. PRECISE LOGARITHMIC ASYMPTOTICS FOR THE RIGHT TAILS OF SOME
LIMIT RANDOM VARIABLES FOR RANDOM TREES
James Allen Fill and Svante Janson
For certain random variables that arise as limits of functionals of
random
finite trees, we obtain precise asymptotics for the logarithm of the
right-hand
tail. Our results are based on the facts (i) that the random
variables we study
can be represented as functionals of a Brownian excursion and (ii)
that a large
deviation principle with good rate function is known explicitly for
Brownian
excursion. Examples include limit distributions of the total path
length and of
the Wiener index in conditioned Galton-Watson trees (also known as
simply
generated trees). In the case of Wiener index (where we recover
results proved
by Svante Janson and Philippe Chassaing by a different method) and
for some
other examples, a key constant is expressed as the solution to a certain
optimization problem, but the constant's precise value remains unknown.
http://front.math.ucdavis.edu/math.PR/0701259
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5111. FLUCTUATIONS OF LEVY PROCESSES AND SCATTERING THEORY
Sonia Fourati
We establish a connection between the scattering inverse problem and the
determination of the distribution of the position of the Levy process
at the
exit time of a bounded interval in term of its Levy exponent.
http://front.math.ucdavis.edu/math.PR/0701271
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5112. AN IMPROVEMENT OF A RESULT ON SMOLYANOV-WEIZSAECKER SURFACE
MEASURES
Evelina Shamarova
Let $M$ be a compact Riemannian manifold without boundary isometrically
embedded into $\Rnu^m$, $\W^x_{M,t}$ be the distribution of a
Brownian bridge
starting at $x\in M$ and returning to $M$ at time $t$. Let $Q_t: \C
(M) \to
\C(M)$, $(Q_t f)(x)=\int_{\C([0,1],\Rnu^m)}f(\om(t)) \W^x_{M,t}(d\om)
$, and let
$\mc P = \{0=t_0 < t_1 < ... < t_n=t\}$ be a partition of $[0,t]$. It
was shown
in a paper by O. G. Smolyanov, H. v. Weizsaecker, and O. Wittich that
$Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f \to e^{-t\frac{\lap_M}2}f, \text
{as} |\mc
P|\to 0$ in $\C(M)$. Taking into consideration integral representations:
$(Q_{t_1-t_0}... Q_{t_n-t_{n-1}} f)(x)=\int_M q_{_{\mc P}}(x,y)f(y)
\la_M(dy)$
and $(e^{-t\frac{\lap_M}2}f)(x)=\int_M h(x,y,t) f(y) \la_M(dy)$,
where $\la_M$
is the volume measure on $M$, $h(x,y,t)$ is the heat kernel on $M$, one
interprets this relation as a weak convergence in $\C(M)$ of the
integral
kernels: $q_{\mc P}(x,y)\to h(x,y,t)$. The present paper improves the
result by
Smolyanov and Weizsaecker, and shows that this convergence is uniform
on $M\x
M$. Keywords: Gaussian integrals on compact Riemannian manifolds,
heat kernel,
Smolyanov--Weizsaecker approach, Smolyanov--Weizsaecker surface measures
http://front.math.ucdavis.edu/math.PR/0701281
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5113. THE CONVERGENCE TO EQUILIBRIUM OF NEUTRAL GENETIC MODELS
Pierre Del Moral (JAD and IRISA / INRIA Rennes) and Laurent Miclo
(LATP) and Fr\'{e}d\'{e}ric Patras (JAD), Sylvain Rubenthaler (JAD)
This article is concerned with the long time behavior of neutral genetic
population models, with fixed population size. We design an explicit,
finite,
exact, genealogical tree based representation of stationary
populations that
holds both for finite and infinite types (or alleles) models. We then
analyze
the decays to the equilibrium of finite populations in terms of the
convergence
to stationarity of their first common ancestor. We estimate the Lyapunov
exponent of the distribution flows with respect to the total
variation norm. We
give bounds on these exponents only depending on the stability with
respect to
mutation of a single individual; they are inversely proportional to the
population size parameter.
http://front.math.ucdavis.edu/math.PR/0701284
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5114. SORTING USING COMPLETE SUBINTERVALS AND THE MAXIMUM NUMBER OF
RUNS IN A RANDOMLY EVOLVING SEQUENCE
Svante Janson
We study the space requirements of a sorting algorithm where only
items that
at the end will be adjacent are kept together. This is equivalent to the
following combinatorial problem: Consider a string of fixed length n
that
starts as a string of 0's, and then evolves by changing each 0 to 1,
with then
changes done in random order. What is the maximal number of runs of 1's?
We give asymptotic results for the distribution and mean. It turns
out that,
as in many problems involving a maximum, the maximum is
asymptotically normal,
with fluctuations of order n^{1/2}, and to the first order well
approximated by
the number of runs at the instance when the expectation is maximized,
in this
case when half the elements have changed to 1; there is also a second
order
term of order n^{1/3}.
We also treat some variations, including priority queues. The
proofs use
methods originally developed for random graphs.
http://front.math.ucdavis.edu/math.PR/0701288
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5115. CRITICAL RANDOM GRAPHS: DIAMETER AND MIXING TIME
Asaf Nachmias and Yuval Peres
Let C_1 denote the largest connected component of the critical Erdos-
Renyi
random graph G(n,1/n). We show that, typically, the diameter of C_1
is of order
n^{1/3} and the mixing time of the lazy simple random walk on C_1 is
of order
n. The latter answers a question of Benjamini, Kozma and Wormald.
These results
extend to clusters of size n^{2/3} of p-bond percolation on any d-
regular
n-vertex graph where such clusters exist, provided that p(d-1) \leq 1.
http://front.math.ucdavis.edu/math.PR/0701316
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5116. ASYMPTOTIC NORMALITY FOR TRACES OF POLYNOMIALS IN INDEPENDENT
COMPLEX WISHART MATRICES
Wlodek Bryc
We derive a non-asymptotic expression for the moments of traces of
monomials
in several independent complex Wishart matrices, extending some explicit
formulas available in the literature. We then deduce the explicit
expression
for the cumulants. From the latter, we read out the multivariate normal
approximation to the traces of finite families of polynomials in
independent
complex Wishart matrices.
http://front.math.ucdavis.edu/math.PR/0701318
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5117. SOME THEORY FOR THE ANALYSIS OF RANDOM FIELDS - WITH
APPLICATIONS TO GEOSTATISTICS
Philipp Pluch
MSc thesis written under the supervision of Dr. J. Pilz (Klagenfurt
University) and Dr. W. Mueller (Linz University) during the FWF Project
'Optimal design of correlated random fields'.
http://front.math.ucdavis.edu/math.ST/0701323
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5118. RANDOM MATRICES, THE ULAM PROBLEM, DIRECTED POLYMERS & GROWTH
MODELS, AND SEQUENCE MATCHING
Satya N. Majumdar
In these lecture notes I will give a pedagogical introduction to some
common
aspects of 4 different problems: (i) random matrices (ii) the longest
increasing subsequence problem (also known as the Ulam problem) (iii)
directed
polymers in random medium and growth models in (1+1) dimensions and
(iv) a
problem on the alignment of a pair of random sequences. Each of these
problems
is almost entirely a sub-field by itself and here I will discuss only
some
specific aspects of each of them. These 4 problems have been studied
almost
independently for the past few decades, but only over the last few
years a
common thread was found to link all of them. In particular all of
them share
one common limiting probability distribution known as the Tracy-Widom
distribution that describes the asymptotic probability distribution
of the
largest eigenvalue of a random matrix. I will mention here, without
mathematical derivation, some of the beautiful results discovered in
the past
few years. Then, I will consider two specific models (a) a ballistic
deposition
growth model and (b) a model of sequence alignment known as the
Bernoulli
matching model and discuss, in some detail, how one derives exactly the
Tracy-Widom law in these models. The emphasis of these lectures would
be on how
to map one model to another. Some open problems are discussed at the
end.
http://front.math.ucdavis.edu/cond-mat/0701193
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5119. PERCOLATION ON DENSE GRAPH SEQUENCES
B. Bollobas and C. Borgs and J. Chayes and O. Riordan
In this paper, we determine the percolation threshold for an arbitrary
sequence of dense graphs $(G_n)$. Let $\lambda_n$ be the largest
eigenvalue of
the adjacency matrix of $G_n$, and let $G_n(p_n)$ be the random
subgraph of
$G_n$ that is obtained by keeping each edge independently with
probability
$p_n$. We show that the appearance of a giant component in $G_n(p_n)$
has a
sharp threshold at $p_n=1/\lambda_n$. In fact, we prove much more,
that if
$(G_n)$ converges to an irreducible limit, then the density of the
largest
component of $G_n(c/n)$ tends to the survival probability of a multi-
type
branching process defined in terms of this limit. Here the notions of
convergence and limit are those of Borgs, Chayes, Lov\'asz, S\'os and
Vesztergombi.
In addition to using basic properties of convergence, we make
heavy use of
the methods of Bollob\'as, Janson and Riordan, who used such branching
processes to study the emergence of a giant component in a very broad
family of
sparse inhomogeneous random graphs.
http://front.math.ucdavis.edu/math.PR/0701346
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5120. A GRAPH THEORETIC INTERPRETATION OF THE MEAN FIRST PASSAGE TIMES
Pavel Chebotarev
Let $m_{ij}$ be the mean first passage time from state $i$ to state $j
$ in an
$n$-state ergodic homogeneous Markov chain with transition matrix $T
$. Let $G$
be the weighted digraph without loops whose vertex set is the set of
states of
the Markov chain and arc weights are equal to the corresponding
transition
probabilities. We give a graph-theoretic interpretation to $m_{ij}$.
Namely, we
show that $m_{ij}=1/q_j$ if $i=j$ and $m_{ij}=f_{ij}/(\sigma q_j)$ if
$i\ne j$,
where $f_{ij}$ is the total weight of 2-tree spanning converging
forests in $G$
that have one tree containing $i$ and the other tree converging to $j
$, $q_j$
is the total weight of spanning trees converging to $j$, and $\sigma$
is the
total weight of spanning converging trees in $G$.
http://front.math.ucdavis.edu/math.PR/0701359
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5121. EFFICIENT ESTIMATION OF THE CARDINALITY OF LARGE DATA SETS
Philippe Chassaing (IECN) and Lucas Gerin (IECN)
F.Giroire has recently proposed an algorithm which returns the
approximate
number of distincts elements in a large sequence of words, under strong
constraints coming from the analysis of large data bases. His
estimation is
based on statistical properties of uniform random variables in $[0,1]
$. In this
note we propose an optimal estimation, using Kullback information and
estimation theory.
http://front.math.ucdavis.edu/math.ST/0701347
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5122. JARZYNSKI'S IDENTITY
Evelina Shamarova
Jarzynski's identity (non-equilibrium work theorem) relates the
equilibrium
free energy difference $\Dl F$ to the work $W$ carried out on a
system during a
non-equilibrium transformation. In physics literature, the identity
is usually
written in the form: $ e^{-\beta W} = e^{-\beta\Dl F}$, where the
average is
said to be taken over all trajectories in the phase space. The
identity in this
form has been derived in different ways and published by many
authors. Since
the identity contains the "average over trajectories", it is natural to
interpret this average as the expectation relative to a probability
measure on
trajectories, while assuming that the system evolves stochastically.
In the
present work, Jarzynski's identity is formulated and proved
mathematically
rigorous. It is written in the form $\mathbb E[e^{-\beta W}] = e^{-
\beta\Dl
F}$, where $\mathbb E$ is the expectation relative to a probability
measure on
phase space paths. For this probability measure, some analytical
assumptions
under which Jarzynki's identity holds, are found. Keywords: Probability
measures on phase space paths, integration over phase space paths,
non-equilibrium statistical mechanics, rigorous consideration of
Jarzynski's
identity
http://front.math.ucdavis.edu/math.PR/0701360
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5123. A PARTICLE SYSTEM IN INTERACTION WITH A RAPIDLY VARYING
ENVIRONMENT: MEAN FIELD LIMITS AND APPLICATIONS
Charles Bordenave and David McDonald and Alexandre Proutiere
We study an interacting particle system whose dynamics depends on an
interacting random environment. As the number of particles grows
large, the
transition rate of the particles slows down (perhaps because they
share a
common resource of fixed capacity). The transition rate of a particle is
determined by its state, by the empirical distribution of all the
particles and
by a rapidly varying environment. The transitions of the environment are
determined by the empirical distribution of the particles. We prove the
propagation of chaos on the path space of the particles and establish
that the
limiting trajectory of the empirical measure of the states of the
particles
satisfies a deterministic differential equation. This deterministic
differential equation involves the time averages of the environment
process.
We apply our results to analyze the performance of communication
networks
where users access some resources using random distributed multi-access
algorithms. For these networks, we show that the environment process
corresponds to a process describing the number of clients in a
certain loss
network, which allows us provide simple and explicit expressions of
the network
performance.
http://front.math.ucdavis.edu/math.PR/0701363
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5124. ON UNIQUENESS OF MAXIMAL COUPLING FOR DIFFUSION PROCESSES WITH
A REFLECTION
Kazumasa Kuwada
A maximal coupling of two diffusion processes makes two diffusion
particles
meet as early as possible. We study the uniqueness of maximal
couplings under a
sort of "reflection structure" which ensures the existence of such
couplings.
In this framework, the uniqueness in the class of Markovian couplings
holds for
the Brownian motion on a Riemannian manifold whereas it fails in more
singular
cases. We also prove that a Kendall-Cranston coupling is maximal
under the
reflection structure.
http://front.math.ucdavis.edu/math.PR/0701372
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5125. THE BIRTHDAY PROBLEM AND MARKOV CHAIN MONTE CARLO
Itai Benjamini and Ben Morris
We study the problem of generating a sample from the stationary
distribution
of a Markov chain, given a method to simulate the chain. We give an
approximation algorithm for the case of a random walk on a regular
graph with n
vertices that runs in expected time O^*(\sqrt{n} x L^2-mixing time).
This is
close to the best possible, since \sqrt{n} is a lower bound on the
worst-case
expected running time of any algorithm.
http://front.math.ucdavis.edu/math.PR/0701390
---------------------------------------------------------------
5126. MULTIVARIATE REGULAR VARIATION OF HEAVY-TAILED MARKOV CHAINS
Johan Segers
The upper extremes of a Markov chain with regulary varying stationary
marginal distribution are known to exhibit under general conditions a
multiplicative random walk structure called the tail chain. More
generally, if
the Markov chain is allowed to switch from positive to negative
extremes or
vice versa, the distribution of the tail chain increment may depend
on the sign
of the tail chain on the previous step. But even then, the forward
and backward
tail chain mutually determine each other through a kind of adjoint
relation. As
a consequence, the finite-dimensional distributions of the Markov
chain are
multivariate regularly varying in a way determined by the back-and-
forth tail
chain. An application of the theory yields the asymptotic
distribution of the
past and the future of the solution to a stochastic difference equation
conditionally on the present value being large in absolute value.
http://front.math.ucdavis.edu/math.PR/0701411
---------------------------------------------------------------
5127. HYDRODYNAMICS FOR A NON-CONSERVATIVE INTERACTING PARTICLE SYSTEM
Glauco Valle
We study a simple one-dimensional model which is roughly based on the
spread
of rainfall on a volume already occupied by a incompressible fluid
aiming to
describe the microscopic evolution of the density of mass of the
fluid in
infinite volume under local regular increase of mass of the system
and obtain
the macroscopic behaviour through the hydrodynamic limit.
http://front.math.ucdavis.edu/math.PR/0701413
---------------------------------------------------------------
5128. A LOWER BOUND ON THE DISCONNECTION TIME OF A DISCRETE CYLINDER
Amir Dembo and Alain-Sol Sznitman
We study the asymptotic behavior for large N of the disconnection
time T_N of
simple random walk on a discrete cylinder with base a d-dimensional
discrete
torus of side-length N. When d is sufficiently large, we are able to
substantially improve the lower bounds obtained by the authors in a
previous
article when d is bigger or equal to 2. We show here that the laws of
N^(2d)/T_N are tight.
http://front.math.ucdavis.edu/math.PR/0701414
---------------------------------------------------------------
5129. A PHASE TRANSITION FOR COMPETITION INTERFACES
Pablo A. Ferrari and James B. Martin and Leandro P. R. Pimentel
We study the competition interface between two growing clusters in a
growth
model associated to last-passage percolation. When the initial
unoccupied set
is approximately a cone, we show that this interface has an asymptotic
direction with probability 1. The behaviour of this direction depends
on the
angle theta of the cone: for theta greater or equal to 180, the
direction is
deterministic, while for theta smaller than 180, it is random, and its
distribution can be given explicitly in certain cases. We also obtain
partial
results on the fluctuations of the interface around its asymptotic
direction.
The evolution of the competition interface in the growth model can be
mapped
onto the path of a second-class particle in the totally asymmetric
simple
exclusion process; from the existence of the limiting direction for the
interface, we obtain a new and rather natural proof of the strong law
of large
numbers (with perhaps a random limit) for the position of the second-
class
particle at large times.
http://front.math.ucdavis.edu/math.PR/0701418
---------------------------------------------------------------
5130. TAIL ASYMPTOTICS FOR DISCRETE EVENT SYSTEMS
Marc Lelarge
In the context of communication networks, the framework of stochastic
event
graphs allows a modeling of control mechanisms induced by the
communication
protocol and an analysis of its performances. We concentrate on the
logarithmic
tail asymptotics of the stationary response time for a class of
networks that
admit a representation as (max,plus)-linear systems in a random
medium. We are
able to derive analytic results when the distribution of the holding
times are
light-tailed. We show that the lack of independence may lead in
dimension
bigger than one to non-trivial effects in the asymptotics of the
sojourn time.
We also study in detail a simple queueing network with multipath
routing.
http://front.math.ucdavis.edu/math.PR/0701420
---------------------------------------------------------------
5131. A FRACTIONAL GENERALIZATION OF THE POISSON PROCESSES
Francesco Mainardi and Rudolf Gorenflo and Enrico Scalas
It is our intention to provide via fractional calculus a
generalization of
the pure and compound Poisson processes, which are known to play a
fundamental
role in renewal theory, without and with reward, respectively. We
first recall
the basic renewal theory including its fundamental concepts like
waiting time
between events, the survival probability, the counting function. If
the waiting
time is exponentially distributed we have a Poisson process, which is
Markovian. However, other waiting time distributions are also
relevant in
applications, in particular such ones with a fat tail caused by a
power law
decay of its density. In this context we analyze a non-Markovian renewal
process with a waiting time distribution described by the Mittag-Leffler
function. This distribution, containing the exponential as particular
case, is
shown to play a fundamental role in the infinite thinning procedure of a
generic renewal process governed by a power asymptotic waiting time.
We then
consider the renewal theory with reward that implies a random walk
subordinated
to a renewal process.
http://front.math.ucdavis.edu/math.PR/0701454
---------------------------------------------------------------
5132. RENEWAL PROCESSES OF MITTAG-LEFFLER AND WRIGHT TYPE
Francesco Mainardi and Rudolf Gorenflo and Alessandro Vivoli
After sketching the basic principles of renewal theory we discuss the
classical Poisson process and offer two other processes, namely the
renewal
process of Mittag-Leffler type and the renewal process of Wright
type, so named
by us because special functions of Mittag-Leffler and of Wright type
appear in
the definition of the relevant waiting times. We compare these three
processes
with each other, furthermore consider corresponding renewal processes
with
reward and numerically their long-time behaviour.
http://front.math.ucdavis.edu/math.PR/0701455
---------------------------------------------------------------
5133. OPTIMAL CONTROL OF A LARGE DAM, TAKING INTO ACCOUNT THE WATER
COSTS
Vyacheslav M. Abramov
Consider a large dam model, which is characterized by an upper level
$L^{upper}$ and lower level $L^{lower}$, and if in time $t$ the level
of water
$L_t$ is between these bounds, then the dam is said to be in a normal
state.
The value $L$ = $L^{upper}$ - $L^{lower}$ is assumed to be large. The
passage
of lower or upper bounds leads to damage, the cost per time unit of
which is
$J_1=j_1L$ and $J_2=j_2L$ correspondingly, where $j_1$ and $j_2$ are
given
constants. Let $c_{L_t}$ denote a water cost, depending on the level
of water
in time $t$, $L^{lower}<L_t\leq L^{upper}$. Assuming that
$p_1$=$\lim_{t\to\infty}\mathbf{P}\{L_t=L^{lower}\}$,
$p_2$=$\lim_{t\to\infty}\mathbf{P}\{L_t>L^{upper}\}$ and
$q_i$=$\lim_{t\to\infty}\mathbf{P}\{L_t=i\}$ ($L^{lower}<i\leq L^
{upper}$)
exist, the aim of the paper is to choose the parameters of an output
stream
(specifically defined in the paper) minimizing the long-run expenses
$$J=p_1J_1+p_2J_2+\sum_{i=L^{lower}+1}^{L^{upper}}q_ic_i.$$
The more particular problem, not taking into account the water
costs, has
been recently studied in [Abramov, \emph{J. Appl. Prob.} 44 (2007),
to appear].
The circle of problems discussed in the present paper is related to the
question \textit{How a structure of the water costs affects an optimal
solution?}.
http://front.math.ucdavis.edu/math.PR/0701458
---------------------------------------------------------------
5134. MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS
Sourav Chatterjee and Elizabeth Meckes
Since the introduction of Stein's method in the early 1970s, much
research
has been done in extending and strengthening it; however, there does
not exist
a version of Stein's original method of exchangeable pairs for
multivariate
normal approximation. The aim of this article is to fill this void.
We present
two abstract normal approximation theorems using exchangeable pairs in
multivariate contexts, one for situations in which the underlying
symmetries
are discrete, and one for situations involving continuous symmetry
groups. We
provide several illustrative examples, including a multivariate
version of
Hoeffding's combinatorial central limit theorem and a treatment of
projections
of Haar measure on the orthogonal and unitary groups.
http://front.math.ucdavis.edu/math.PR/0701464
---------------------------------------------------------------
5135. ON CLASSICAL ANALOGUES OF FREE ENTROPY DIMENSION
A. Guionnet and D. Shlyakhtenko
We define a classical probability analogue of Voiculescu's free entropy
dimension that we shall call the classical probability entropy
dimension of a
probability measure on $\mathbb{R}^n$. We show that the classical
probability
entropy dimension of a measure is related with diverse other notions of
dimension. First, it can be viewed as a kind of fractal dimension.
Second, if
one extends Bochner's inequalities to a measure by requiring that
microstates
around this measure asymptotically satisfy the classical Bochner's
inequalities, then we show that the classical probability entropy
dimension
controls the rate of increase of optimal constants in Bochner's
inequality for
a measure regularized by convolution with the Gaussian law as the
regularization is removed. We introduce a free analogue of the Bochner
inequality and study the related free entropy dimension quantity. We
show that
it is greater or equal to the non-microstates free entropy dimension.
http://front.math.ucdavis.edu/math.PR/0701465
---------------------------------------------------------------
5136. ON THE HARDNESS OF SAMPLING INDEPENDENT SETS BEYOND THE TREE
THRESHOLD
Elchanan Mossel and Dror Weitz and Nicholas Wormald
We consider local Markov chain Monte-Carlo algorithms for sampling
from the
weighted distribution of independent sets with activity $\l$, where
the weight
of an independent set $I$ is $\l^{|I|}$. A recent result has
established that
Gibbs sampling is rapidly mixing in sampling the distribution for
graphs of
maximum degree $d$ and $\l<\l_c(d)$, where $\l_c(d)$ is the critical
activity
for uniqueness of the Gibbs measure (i.e., for decay of correlations
with
distance in the weighted distribution over independent sets) on the $d
$-regular
infinite tree.
We show that for $d \geq 3$, $\l$ just above $\l_c(d)$ with high
probability
over $d$-regular bipartite graphs, any local Markov chain Monte-Carlo
algorithm
takes exponential time before getting close to the stationary
distribution.
Our results provide a rigorous justification for ``replica'' method
heuristics. These heuristics were invented in theoretical physics and
are used
in order to derive predictions on Gibbs measures on random graphs in
terms of
Gibbs measures on trees. We conjecture that $\l_c$ is in fact the exact
threshold for this computational problem, i.e., that for $\l>\l_c$ it is
NP-hard to approximate the above weighted sum overindependent sets to
within a
factor polynomial in the size of the graph.
http://front.math.ucdavis.edu/math.PR/0701471
---------------------------------------------------------------
5137. THE EVOLUTION OF THE MIXING RATE
Nikolaos Fountoulakis and Bruce Reed
In this paper we present a study of the mixing time of a random walk
on the
largest component of a supercritical random graph, also known as the
giant
component. We identify local obstructions that slow down the random
walk, when
the average degree d is at most (ln n lnln n)^{1/2}, proving that the
mixing
time in this case is O((ln n/d)^2) asymptotically almost surely. As
the average
degree grows these become negligible and it is the diameter of the
largest
component that takes over, yielding mixing time O(ln n/ln d). We
proved these
results during the 2003-04 academic year. Similar results but for
constant d
were later proved independently by I. Benjamini, G. Kozma and N.
Wormald.
http://front.math.ucdavis.edu/math.CO/0701474
---------------------------------------------------------------
5138. THE GHOSTS OF THE ECOLE NORMALE. LIFE, DEATH AND DESTINY OF REN
\'{E} GATEAUX
Laurent Mazliak (PMA and IMJ)
The present paper deals with the life and some aspects of the scientific
contribution of the mathematician Ren\'{e} Gateaux, killed during
World War 1
at the age of 25. Though he died very young, he left interesting
results in
functional analysis. In particular, he was among the first to try to
construct
an integral over an infinite dimensional space. His ideas were
extensively
developed later by L\'{e}vy. Among other things, he interpreted
Gateaux's
integral in a probabilistic framework that later led to the
construction of
Wiener measure. This article tries to explain this singular personal and
professional destiny in pre and postwar France. It also recalls the
slaughter
inflicted on French students during the conflict.
http://front.math.ucdavis.edu/math.HO/0701490
---------------------------------------------------------------
5139. COMPUTATION TREE AND STRONG SPATIAL MIXING IN MULTI-SPIN SYSTEMS
Chandra Nair and Prasad Tetali
This paper deals with the construction of a computation tree
(hypertree) for
interacting systems modeled using graphs (hypergraphs) that preserve the
marginal probability of any vertex of interest. Local message passing
equations
have been used for some time to approximate the marginal
probabilities in
graphs but it is known that these equations are incorrect for graphs
with
loops. In this paper we construct, for any finite graph and a fixed
vertex, a
finite computation tree with appropriately defined boundary
conditions so that
the marginal probability on the tree at the vertex matches that on
the graph.
For several interacting systems, we show using our approach that if
there is
strong spatial mixing on an infinite regular tree, then one has
strong spatial
mixing for any given graph with maximum degree bounded by that of the
regular
tree. Thus we identify the regular tree as the worst case graph for
the notion
of strong spatial mixing.
http://front.math.ucdavis.edu/math.PR/0701494
---------------------------------------------------------------
5140. ON SINGULAR INTEGRAL AND MARTINGALE TRANSFORMS
S. Geiss and S. Montgomery-Smith and E. Saksman
Linear equivalences of norms of vector-valued singular integral
operators and
vector-valued martingale transforms are studied. In particular, it is
shown
that the UMD(p)-constant of a Banach space X equals the norm of the
real (or
the imaginary) part of the Beurling-Ahlfors singular integral
operator, acting
on the X-valued L^p-space on the plane. Moreover, replacing equality
by a
linear equivalence, this is found to be the typical property of even
multipliers. A corresponding result for odd multipliers and the Hilbert
transform is given.
http://front.math.ucdavis.edu/math.CA/0701516
---------------------------------------------------------------
5141. PENALIZATIONS OF THE BROWNIAN MOTION BY A FUNCTIONAL OF ITS
LOCAL TIMES
Joseph Najnudel
In this article, we study the family of probability measures (indexed
by a
positive real number t), obtained by penalization of the Brownian
motion by a
given functional of its local times at time t. We prove that this
family tends
to a limit measure when t goes to infinity if the functional
satisfies some
conditions of domination, and we check these conditions in several
particular
cases.
http://front.math.ucdavis.edu/math.PR/0701526
---------------------------------------------------------------
5142. VOTER MODELS WITH HETEROZYGOSITY SELECTION
Anja Sturm and Jan Swart
This paper studies variations of the usual voter model that favour
types that
are locally less common. Such models are dual to certain systems of
branching
annihilating random walks that are parity preserving. For both the
voter models
and their dual branching annihilating systems we determine all
homogeneous
invariant laws, and we study convergence to these laws started from
other
initial laws.
http://front.math.ucdavis.edu/math.PR/0701555
---------------------------------------------------------------
5143. HARMONIC ANALYSIS ON A FINITE HOMOGENEOUS SPACE
Fabio Scarabotti and Filippo Tolli
In this paper, we study harmonic analysis on finite homogeneous
spaces whose
associated permutation representation decomposes with multiplicity.
After a
careful look at Frobenius reciprocity and transitivity of induction,
and the
introduction of three types of spherical functions, we develop a
theory of
Gelfand Tsetlin bases for permutation representations. Then we study
several
concrete examples on the symmetric groups, generalizing the Gelfand
pair of the
Johnson scheme; we also consider statistical and probabilistic
applications.
After that, we consider the composition of two permutation
representations,
giving a non commutative generalization of the Gelfand pair
associated to the
ultrametric space; actually, we study the more general notion of crested
product. Finally, we consider the exponentiation action, generalizing
the
decomposition of the Gelfand pair of the Hamming scheme; actually, we
study a
more general construction that we call wreath product of permutation
representations, suggested by the study of finite lamplighter random
walks. We
give several examples of concrete decompositions of permutation
representations
and several explicit 'rules' of decomposition.
http://front.math.ucdavis.edu/math.RT/0701533
---------------------------------------------------------------
5144. NETWORKS OF RECURRENT EVENTS, A THEORY OF RECORDS, AND AN
APPLICATION TO FINDING CAUSAL SIGNATURES IN SEISMICITY
J. Davidsen and P. Grassberger and M. Paczuski
We propose a method to search for signs of causal structure in
spatiotemporal
data making minimal a priori assumptions about the underlying
dynamics. To this
end, we generalize the elementary concept of recurrence for a point
process in
time to recurrent events in space and time. An event is defined to be a
recurrence of any previous event if it is closer to it in space than
all the
intervening events. As such, each sequence of recurrences for a given
event is
a record breaking process. This definition provides a strictly data
driven
technique to search for structure. Defining events to be nodes, and
linking
each event to its recurrences, generates a network of recurrent events.
Significant deviations in properties of that network compared to
networks
arising from random processes allows one to infer attributes of the
causal
dynamics that generate observable correlations in the patterns. We
derive
analytically a number of properties for the network of recurrent events
composed by a random process. We extend the theory of records to
treat not only
the variable where records happen, but also time as continuous. In
this way, we
construct a fully symmetric theory of records leading to a number of new
results. Those analytic results are compared to the properties of a
network
synthesized from earthquakes in Southern California. Significant
disparities
from the ensemble of acausal networks that can be plausibly
attributed to the
causal structure of seismicity are: (1) Invariance of network
statistics with
the time span of the events considered, (2) Appearance of a
fundamental length
scale for recurrences, independent of the time span of the catalog,
which is
consistent with observations of the ``rupture length'', (3) Hierarchy
in the
distances and times of subsequent recurrences.
http://front.math.ucdavis.edu/physics/0701190
---------------------------------------------------------------
5145. EXIT ASYMPTOTICS FOR SMALL DIFFUSION ABOUT AN UNSTABLE EQUILIBRIUM
Yuri Bakhtin
A dynamical system perturbed by white noise in a neighborhood of an
unstable
fixed point is considered. We obtain the exit asymptotics in the
limit of
vanishing noise intensity. This is a refinement of a result by Kifer
(1981).
http://front.math.ucdavis.edu/math.PR/0701569
---------------------------------------------------------------
5146. GENERATING RANDOM VECTORS IN (Z/PZ)^D VIA AN AFFINE RANDOM PROCESS
Martin Hildebrand and Joseph McCollum
This paper considers some random processes of the form X_{n+1}=TX_n
+B_n (mod
p) where B_n and X_n are random variables over (Z/pZ)^d and T is a
fixed d x d
integer matrix which is invertible over the complex numbers. For a
particular
distribution for B_n, this paper improves results of Asci to show
that if T has
no complex eigenvalues of length 1, then for integers p relatively
prime to
det(T), order (log p)^2 steps suffice to make X_n close to uniformly
distributed where X_0 is the zero vector. This paper also shows that
if T has a
complex eigenvalue which is a root of unity, then order p^b steps are
needed
for X_n to get close to uniform where b is a value which may depend
on T and
X_0 is the zero vector.
http://front.math.ucdavis.edu/math.PR/0701570
---------------------------------------------------------------
5147. MEINARDUS' THEOREM ON WEIGHTED PARTITIONS: EXTENSIONS AND A
PROBABILISTIC PROOF
Boris L. Granovsky and Dudley Stark and Michael Erlihson
We give a probalistic proof of the famous Meinardus' asymptotic
formula for
the number of weighted partitions with weakened one of the three
Meinardus'
conditions, and extend the resulting version of the theorem to other two
classis types of decomposable combinatorial structures, which are called
assemblies and selections. The results obtained are based on combining
Meinardus' analytical approach with probabilistic method of Khitchine.
http://front.math.ucdavis.edu/math.PR/0701584
---------------------------------------------------------------
5148. HARMONIC ANALYSIS OF FINITE LAMPLIGHTER RANDOM WALKS
Fabio Scarabotti and Filippo Tolli
Recently, several papers have been devoted to the analysis of
lamplighter
random walks, in particular when the underlying graph is the infinite
path
$\mathbb{Z}$. In the present paper, we develop a spectral analysis for
lamplighter random walks on finite graphs. In the general case, we
use the
$C_2$-symmetry to reduce the spectral computations to a series of
eigenvalue
problems on the underlying graph. In the case the graph has a transitive
isometry group $G$, we also describe the spectral analysis in terms
of the
representation theory of the wreath product $C_2\wr G$. We apply our
theory to
the lamplighter random walks on the complete graph and on the
discrete circle.
These examples were already studied by Haggstrom and Jonasson by
probabilistic
methods.
http://front.math.ucdavis.edu/math.PR/0701603
---------------------------------------------------------------
5149. CONNECTED ALLOCATION TO POISSON POINTS IN R^2
Maxim Krikun (IECN)
his note answers one question in [math.PR/0505668], concerning the
connected
allocation for the Poisson process in R^2. The proposed solution
makes use of
the Riemann map from the plane minus the minimal spanning forest of
the Poisson
point process to the halfplane. A picture of a numerically simulated
example is
included.
http://front.math.ucdavis.edu/math.PR/0701611
---------------------------------------------------------------
5150. ASYMPTOTIC ENUMERATION OF DENSE 0-1 MATRICES WITH SPECIFIED
LINE SUMS AND FORBIDDEN POSITIONS
Catherine Greenhill and Brendan D. McKay
Let S=(s_1,s_2,..., s_m) and T = (t_1,t_2,..., t_n) be vectors of
non-negative integers with \sum_{i=1}^m s_i = \sum_{j=1}^n t_j, and let
X=(x_{jk}) be an m*n matrix over {0,1}. Define B(S,T,X) to be the
number of m*n
matrices B=(b_{jk}) over {0,1} with row sums given by S and column
sums given
by T such that x_{jk}=1 implies b_{jk}=0 for all j,k. That is, X
specifies a
set of entries of B required to be 0. Equivalently, B(S,T,X) is the
number of
bipartite graphs with m vertices in one part with degrees given by S,
and n
vertices in the other part with degrees given by T, and avoiding all
the edges
specified in X. Note that B(S,T,X)/B(S,T,0) is the probability that a
uniformly
chosen {0,1}-matrix with row sums S and column sums T has zeros in
the places
where X is nonzero. An asymptotic formula for B(S,T,X) was given by
McKay
(1984) in the case that the matrices are sparse. In the case of dense
matrices
there seem to be no prior results except for the special case X=0
studied by
Canfield, Greenhill and McKay (math.CO/0606496). This paper extends the
analytic methods used by the latter paper to obtain an asymptotic
formula for
B(S,T,X) in the dense regime where the entries of S and T can vary
within
certain limits and the row and column sums of X are not too large. As
applications, we find the asymptotic number of simple digraphs with
given
vectors of in-degree and out-degree, and the expected permanent of a
{0,1}-matrix with given row and column sums, with both results
holding in the
dense regime.
http://front.math.ucdavis.edu/math.CO/0701600
---------------------------------------------------------------
5151. A LIMIT THEOREM FOR DIFFUSIONS ON GRAPHS WITH VARIABLE
CONFIGURATION
Alexey M. Kulik
A limit theorem for a sequence of diffusion processes on graphs is
proved in
a case when vary both parameters of the processes (the drift and
diffusion
coefficients on every edge and the asymmetry coefficients in every
vertex), and
configuration of graphs, where the processes are set on. The explicit
formulae
for the parameters of asymmetry for the vertices of the limiting
graph are
given in the case, when, in the pre-limiting graphs, some groups of
vertices
form knots contracting into a points.
http://front.math.ucdavis.edu/math.PR/0701632
---------------------------------------------------------------
5152. GROWTH OF PREFERENTIAL ATTACHMENT RANDOM GRAPHS VIA CONTINUOUS-
TIME BRANCHING PROCESSES
K.B. Athreya and A.P. Ghosh and S. Sethuraman
A version of ``preferential attachment'' random graphs, corresponding to
linear ``weights'' with random ``edge additions,'' which generalizes
some
previously considered models, is studied. This graph model is
embedded in a
continuous-time branching scheme and, using the branching process
apparatus,
several results on the graph model asymptotics are obtained, some
extending
previous results, such as growth rates for a typical degree and the
maximal
degree, behavior of the vertex where the maximal degree is attained,
and a law
of large numbers for the empirical distribution of degrees which
shows certain
``scale-free'' or ``power-law'' behaviors.
http://front.math.ucdavis.edu/math.PR/0701649
---------------------------------------------------------------
5153. THE LOWER TAIL PROBLEM FOR HOMOGENEOUS FUNCTIONALS OF STABLE
PROCESSES WITH NO NEGATIVE JUMPS
Thomas Simon (DP)
Let Z be a strictly a-stable real Levy process (a>1) and X be a
fluctuating
b-homogeneous additive functional of Z. We investigate the
asymptotics of the
first passage-time of X above 1, and give a general upper bound. When
Z has no
negative jumps, we prove that this bound is optimal and does not
depend on the
homogeneity parameter b. This extends a result of Y. Isozaki.
http://front.math.ucdavis.edu/math.PR/0701653
---------------------------------------------------------------
5154. SPLITTING PAIRS AND THE NUMBER OF CLUSTERS GENERATED BY RANDOM
PAIR INCOMPATIBILITIES
Damien Pitman
We consider a random fitness landscape on the space of haploid diallelic
genotypes with n genetic loci, where each genotype is considered either
inviable or viable depending on whether or not there are any
incompatibilities
among its allele pairs. We suppose that each allele pair in the set
of all
possible allele pairs on the n loci is independently incompatible with
probability p=c/(2n). We examine the connectivity of the viable
genotypes under
single locus mutations and show that, for 0<c<1, the the number of
clusters of
viable genotypes in this landscape converges weakly (in n) to N=2^
{Psi} where
Psi is Poisson distributed; while for c>1, there are no viable
genotypes with
probability converging to one. The genotype space is equivalent to the
n-dimensional hypercube and the viable genotypes are solutions to a
random
2-SAT problem, so the same result holds for the connectivity of
solutions in
the hypercube to a random 2-SAT problem.
http://front.math.ucdavis.edu/math.PR/0701656
---------------------------------------------------------------
5155. NORMAL FORM TRANSFORMS SEPARATE SLOW AND FAST MODES IN
STOCHASTIC DYNAMICAL SYSTEMS
A. J. Roberts
Modelling stochastic systems has many important applications. Normal
form
coordinate transforms are a powerful way to untangle interesting long
term
macroscale dynamics from detailed microscale dynamics. We explore such
coordinate transforms of stochastic differential systems when the
dynamics has
both slow modes and quickly decaying modes. The thrust is to derive
normal
forms useful for macroscopic modelling of complex stochastic microscopic
systems. Thus we not only must reduce the dimensionality of the
dynamics, but
also endeavour to separate all slow processes from all fast time
processes,
both deterministic and stochastic. Quadratic stochastic effects in
the fast
modes contribute to the drift of the important slow modes. The
results will
help us accurately model, interpret and simulate multiscale
stochastic systems.
http://front.math.ucdavis.edu/math.DS/0701623
---------------------------------------------------------------
5156. ASYMPTOTIC EVOLUTION OF ACYCLIC RANDOM MAPPINGS
Steven N. Evans and Tye Lidman
An acyclic mapping from an $n$ element set into itself is a mapping $
\phi$
such that if $\phi^k(x) = x$ for some $k$ and $x$, then $\phi(x) = x$.
Equivalently, $\phi^\ell = \phi^{\ell+1} = ...$ for $\ell$
sufficiently large.
We investigate the behavior as $n \to \infty$ of a Markov chain on the
collection of such mappings. At each step of the chain, a point in
the $n$
element set is chosen uniformly at random and the current mapping is
modified
by replacing the current image of that point by a new one chosen
independently
and uniformly at random, conditional on the resulting mapping being
again
acyclic. We can represent an acyclic mapping as a directed graph
(such a graph
will be a collection of rooted trees) and think of these directed
graphs as
metric spaces with some extra structure. Heuristic calculations
indicate that
the metric space valued process associated with the Markov chain
should, after
an appropriate time and ``space'' rescaling, converge as $n \to \infty
$ to a
real tree ($\R$-tree) valued Markov process that is reversible with
respect to
a measure induced naturally by the standard reflected Brownian
bridge. The
limit process, which we construct using Dirichlet form methods, is a
Hunt
process with respect to a suitable Gromov-Hausdorff-like metric. This
process
is similar to one that appears in earlier work by Evans and Winter as
the limit
of chains involving the subtree prune and regraft tree (SPR)
rearrangements
from phylogenetics.
http://front.math.ucdavis.edu/math.PR/0701657
---------------------------------------------------------------
5157. DIFFUSIVE VARIANCE FOR A TAGGED PARTICLE IN $D\LEQ 2$
ASYMMETRIC SIMPLE EXCLUSION
Sunder Sethuraman
We study the equilibrium fluctuations of a tagged particle in finite-
range
simple exclusion processes on Z^d with biased single particle jump
rates. It is
known the variance of the tagged particle at time t is diffusive,
that is on
order O(t), in d\geq 3, and in d=1 when in addition the jump rate is
nearest-neighbor, and moreover, in these cases, central limit
theorems in
diffusive scale have been proved.
In this article, we give some partial results in the open cases in
d\leq 2.
Namely, we show diffusivity of the tagged particle variance at time t
in the
sense of some upper and lower bounds on order O(t) in d=2, and also
in d=1 when
in addition the jump rate is not nearest-neighbor. Also, a
characterization of
the tagged particle variance is given. The main methods are in
analyzing H_{-1}
norm variational inequalities.
http://front.math.ucdavis.edu/math.PR/0701660
---------------------------------------------------------------
5158. CRITICAL AGE DEPENDENT BRANCHING MARKOV PROCESSES AND THEIR
SCALING LIMITS
Krishna Athreya and Siva Athreya and and Srikanth Iyer
This paper studies: (i) the long time behaviour of the empirical
distribution
of age and normalised position of an age dependent critical branching
Markov
process conditioned on non-extinction; and (ii) the super-process
limit of a
sequence of age dependent critical branching Brownian motions.
http://front.math.ucdavis.edu/math.PR/0701661
---------------------------------------------------------------
5159. SHAPE CURVATURES AND TRANSVERSAL FLUCTUATIONS IN THE FIRST
PASSAGE PERCOLATION MODEL
Yu Zhang
We consider the first passage percolation model on the square
lattice. In
this model, $\{t(e): e{an edge of}{\bf Z}^2 \}$ is an independent
identically
distributed family with a common distribution $F$. We denote by $T
({\bf 0}, v)$
the passage time from the origin to $v$ for $v\in {\bf R}^2$ and $B(t)
=\{v\in
{\bf R}^d: T({\bf 0}, v)\leq t\}.$ It is well known that if $F(0) <
p_c$, there
exists a compact shape ${\bf B}_F\subset {\bf R}^2$ such that for all
$\epsilon
>0$, $t {\bf B}_F(1-\epsilon) \subset {B(t)} \subset t{\bf B}_F(1+
\epsilon)$,
eventually with a probability 1. For each shape boundary point $u$,
we denote
its right- and left-curvature exponents by $\kappa^+(u)$ and $\kappa^-
(u)$. In
addition, for each vector $u$, we denote the transversal fluctuation
exponent
by $\xi(u)$. In this paper, we can show that $\xi(u) \leq
1-\max\{\kappa^-(u)/2, \kappa^+(u)/2\}$ for all shape boundary points
$u$.
To pursue a curvature on ${\bf B}_F$, we consider passage times
with a
special distribution infsupp$(F)=l$ and $F(l)=p > \vec{p}_c$, where $l
$ is a
positive number and $\vec{p}_c$ is a critical point for the oriented
percolation model. With this distribution, it is known that there is
a flat
segment on the shape boundary between angles $0< \theta_p^- <
\theta_p^+<
90^\circ$. In this paper, we show that the shape are strictly convex
at the
directions $\theta_p^\pm$. Moreover, we also show that for all $r>0$,
$\xi((r,
\theta^\pm_p)) = 0.5$ and $\xi((r, \theta)) =1$ for all $\theta_p^- <
\theta<
\theta_p^+$ and $r>0$.
http://front.math.ucdavis.edu/math.PR/0701689
---------------------------------------------------------------
5160. SPATIAL EPIDEMICS: CRITICAL BEHAVIOR IN ONE DIMENSION
Steven P. Lalley
In the simple mean-field SIS and SIR epidemic models, infection is
transmitted from infectious to susceptible members of a finite
population by
independent p-coin tosses. Spatial variants of these models are
proposed, in
which finite populations of size N are situated at the sites of a
lattice and
infectious contacts are limited to individuals at neighboring sites.
Scaling
laws for these models are given when the infection parameter p is
such that the
epidemics are critical. It is shown that in all cases there is a
critical
threshold for the numbers initially infected: below the threshold,
the epidemic
evolves in essentially the same manner as its branching envelope, but
at the
threshold evolves like a branching process with a size-dependent
drift. The
corresponding scaling limits are super-Brownian motions and Dawson-
Watanabe
processes with killing, respectively.
http://front.math.ucdavis.edu/math.PR/0701698
---------------------------------------------------------------
5161. NOTES ON THE OCCUPANCY PROBLEM WITH INFINITELY MANY BOXES:
GENERAL ASYMPTOTICS AND POWER LAWS
Alexander Gnedin and Ben Hansen and Jim Pitman
This paper collects facts about the number of occupied boxes in the
classical
balls-in-boxes occupancy scheme with infinitely many positive
frequencies:
equivalently, about the number of species represented in samples from
populations with infinitely many species. We present moments of this
random
variable, discuss asymptotic relations among them and with related
random
variables, and draw connections with regular variation, which appears in
various manifestations.
http://front.math.ucdavis.edu/math.PR/0701718
---------------------------------------------------------------
5162. DISTANCE ESTIMATES FOR DEPENDENT THINNINGS OF POINT PROCESSES
WITH DENSITIES
Dominic Schuhmacher
In [Schuhmacher, Electron. J. Probab. 10 (2005), 165--201] estimates
of the
Barbour-Brown distance d_2 between the distribution of a thinned
point process
and the distribution of a Poisson process were derived by combining
discretization with a result based on Stein's method. In the present
article we
concentrate on point processes that have a density with respect to a
Poisson
process. For such processes we can apply a corresponding result directly
without the detour of discretization and thus obtain better and more
natural
bounds not only in d_2 but also in the stronger total variation
metric. We give
applications for thinning by covering with an independent Boolean
model and
"Mat{\'e}rn type I"-thinning of fairly general point processes. These
applications give new insight into the respective models, and either
generalize
or improve earlier results.
http://front.math.ucdavis.edu/math.PR/0701728
---------------------------------------------------------------
5163. NON-EQUILIBRIUM STOCHASTIC DYNAMICS IN CONTINUUM: THE FREE CASE
Y. Kondratiev and E. Lytvynov and M. R\"ockner
We study the problem of identification of a proper state-space for the
stochastic dynamics of free particles in continuum, with their
possible birth
and death. In this dynamics, the motion of each separate particle is
described
by a fixed Markov process $M$ on a Riemannian manifold $X$. The main
problem
arising here is a possible collapse of the system, in the sense that,
though
the initial configuration of particles is locally finite, there could
exist a
compact set in $X$ such that, with probability one, infinitely many
particles
will arrive at this set at some time $t>0$. We assume that $X$ has
infinite
volume and, for each $\alpha\ge1$, we consider the set $\Theta_\alpha
$ of all
infinite configurations in $X$ for which the number of particles in a
compact
set is bounded by a constant times the $\alpha$-th power of the
volume of the
set. We find quite general conditions on the process $M$ which
guarantee that
the corresponding infinite particle process can start at each
configuration
from $\Theta_\alpha$, will never leave $\Theta_\alpha$, and has
cadlag (or,
even, continuous) sample paths in the vague topology. We consider the
following
examples of applications of our results: Brownian motion on the
configuration
space, free Glauber dynamics on the configuration space (or a birth-
and-death
process in $X$), and free Kawasaki dynamics on the configuration
space. We also
show that if $X=\mathbb R^d$, then for a wide class of starting
distributions,
the (non-equilibrium) free Glauber dynamics is a scaling limit of
(non-equilibrium) free Kawasaki dynamics.
http://front.math.ucdavis.edu/math.PR/0701736
---------------------------------------------------------------
5164. SEARCH COST FOR A NEARLY OPTIMAL PATH IN A BINARY TREE
Robin Pemantle
Consider a binary tree, to the vertices of which are assigned
independent
Bernoulli random variables with mean p <= 1/2. How many of these
Bernoullis one
must look at in order to find a path of length n from the root which
maximizes,
up to a factor of 1 - epsilon, the sum of the Bernoullis along the
path? In the
case, p = 1/2 (the critical value for nontriviality), it is shown to
take of
order epsilon^{-1} n steps. In the case p < 1/2, the number of steps
is shown
to be exponential in epsilon^{-1/2}. This last result matches Aldous'
upper
bound for a certain family of subcases.
http://front.math.ucdavis.edu/math.PR/0701741
---------------------------------------------------------------
5165. EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE'S WITH A JUMP NOISE
Alexey M.Kulik
The mild sufficient conditions for exponential ergodicity of a Markov
process, defined as the solution to SDE with a jump noise, are given.
These
conditions include three principal claims: recurrence condition R,
topological
irreducibility condition S and non-degeneracy condition N, the latter
formulated in the terms of a certain random subspace of \Re^m,
associated with
the initial equation. The examples are given, showing that, in
general, none of
three principal claims can be removed without losing ergodicity of
the process.
The key point in the approach, developed in the paper, is that the local
Doeblin condition can be derived from N and S via the stratification
method and
criterium for the convergence in variations of the family of induced
measures
on \Re^m.
http://front.math.ucdavis.edu/math.PR/0701747
---------------------------------------------------------------
5166. DECAY RATES LARGE DEVIATIONS FOR THE PLANAR VOTER MODEL
OCCUPATION TIME
G. Maillard and T. Mountford
We study the decay rate of large deviation probabilities of
occupation times,
up to time $t$, for the voter model $\eta\colon\Z^2\times[0,\infty)\ra
\{0,1\}$
with simple random walk transition kernel, starting from a Bernoulli
product
distribution with density $\rho\in(0,1)$. In \cite{bramcoxgri88},
Bramson, Cox
and Griffeath showed that the decay rate order lies in $[\log(t),
\log^2(t)]$.
In this paper, we establish the true decay rates depending on the
level. We
show that the decay rates are $\log^2(t)$ when the deviation from $
\rho$ is
maximal (i.e., $\eta\equiv 0$ or 1), and $\log(t)$ in all other
situations.
This answers some conjecture in \cite{bramcoxgri88} and confirms
analysis
carried out in \cite{benfrakra96}, \cite{dorgod98} and \cite{howgod98}.
http://front.math.ucdavis.edu/math.PR/0701754
---------------------------------------------------------------
5167. AREA DISTRIBUTION AND SCALING FUNCTION FOR PUNCTURED POLYGONS
Christoph Richard and Iwan Jensen and Anthony J. Guttmann
Punctured polygons are polygons with internal holes which are also
polygons.
The external and internal polygons are of the same type, and they are
mutually
as well as self-avoiding. We rigorously analyse the effect of a
finite number
of punctures on the limiting area distribution in a uniform ensemble,
where
punctured polygons with equal perimeter have the same probability of
occurrence. The results rely on an assumption on the limiting area
distribution
for unpunctured polygons. Our analysis leads to conjectures about the
possible
scaling behaviour of the models.
We also analyse exact enumeration data. For staircase polygons
with punctures
of fixed size, we find exact generating functions for the first few
area-moments. For staircase polygons with punctures of arbitrary size, a
careful numerical analysis yields very accurate estimates for the
area-moments.
Interestingly, we find that the leading correction term for each area-
moment is
proportional to the corresponding area-moment with one less puncture. We
finally analyse corresponding quantities for punctured self-avoiding
polygons
and find agreement with the exact formulas to at least 3--4
significant digits.
http://front.math.ucdavis.edu/math.CO/0701633
---------------------------------------------------------------
5168. PARAMETRIZED STOCHASTIC GRAMMARS FOR RNA SECONDARY STRUCTURE
PREDICTION
Robert S. Maier
We propose a two-level stochastic context-free grammar (SCFG)
architecture
for parametrized stochastic modeling of a family of RNA sequences,
including
their secondary structure. A stochastic model of this type can be
used for
maximum a posteriori estimation of the secondary structure of any new
sequence
in the family. The proposed SCFG architecture models RNA subsequences
comprising paired bases as stochastically weighted Dyck-language
words, i.e.,
as weighted balanced-parenthesis expressions. The length of each run of
unpaired bases, forming a loop or a bulge, is taken to have a phase-type
distribution: that of the hitting time in a finite-state Markov
chain. Without
loss of generality, each such Markov chain can be taken to have a
bounded
complexity. The scheme yields an overall family SCFG with a
manageable number
of parameters.
http://front.math.ucdavis.edu/q-bio.BM/0701036
---------------------------------------------------------------
5169. LEARNING TRIGONOMETRIC POLYNOMIALS FROM RANDOM SAMPLES AND
EXPONENTIAL INEQUALITIES FOR EIGENVALUES OF RANDOM MATRICES
Karlheinz Groechenig and Benedikt M. Poetscher and Holger Rauhut
Motivated by problems arising in random sampling of trigonometric
polynomials, we derive exponential inequalities for the operator norm
of the
difference between the sample second moment matrix $n^{-1}U^*U$ and its
expectation where $U$ is a complex random $n\times D$ matrix with
independent
rows. These results immediately imply deviation inequalities for the
largest
(smallest) eigenvalues of the sample second moment matrix, which in
turn lead
to results on the condition number of the sample second moment
matrix. We also
show that trigonometric polynomials in several variables can be
learned from
$const \cdot D \ln D$ random samples.
http://front.math.ucdavis.edu/math.PR/0701781
---------------------------------------------------------------
5170. OCCUPATION LAWS FOR SOME TIME-NONHOMOGENEOUS MARKOV CHAINS
Zach Dietz and Sunder Sethuraman
We consider finite-state time-nonhomogeneous Markov chains where the
probability of moving from state $i$ to state $j\neq i$ at time $n$ is
$G(i,j)/n^\zeta$ for a ``generator'' matrix $G$ and strength parameter
$\zeta>0$. In these chains, as time grows, the positions are less and
less
likely to change, and so form simple models of age-dependent time-
reinforcing
behaviors. These chains, however, exhibit some different, perhaps
unexpected,
asymptotic occupation laws depending on parameters.
Although on the one hand it is shown that the asymptotic position
converges
to a point-mixture for all $\zeta>0$, on the other hand, the average
position,
when variously $0<\zeta<1$, $\zeta>1$ or $\zeta=1$, is shown to
converges to a
constant, a point-mixture, or a distribution $\mu_G$ with no atoms
and full
support on a certain simplex respectively. The last type of limit can
be seen
as a sort of ``spreading'' between the cases $0<\zeta<1$ and $\zeta>1$.
In particular, when $G$ is appropriately chosen, $\mu_G$ is a
Dirichlet
distribution with certain parameters, reminiscent of results in Polya
urns.
http://front.math.ucdavis.edu/math.PR/0701798
---------------------------------------------------------------
5171. WEAK CONVERGENCE OF STEP PROCESSES AND AN APPLICATION FOR
CRITICAL MULTITYPE BRANCHING PROCESSES WITH IMMIGRATION
M\'arton Isp\'any and Gyula Pap
First, sufficient conditions are given for a system $(U^n_k)_{n\in\NN,
k\in\ZZ_+}$ of random variables in $\RR^d$ and for a diffusion process
$(\cU_t)_{t\in\RR_+}$ such that $\cU^n\distr\cU$, where
$\cU^n_t:=\sum_{k=0}^{\nt}U^n_k$. Next, sufficient conditions are
given for a
system $(\psi_{n,k})_{n\in\NN, k\in\ZZ_+}$ of Borel functions
$\psi_{n,k}:(\RR^d)^{k+1}\to\RR^p$ and for a measurable mapping
$\Psi:\DD(\RR^d)\to\DD(\RR^p)$ such that
$(\cU^n,\cV^n,\cY^n)\distr(\cU,\cV,\cY)$, where $\cV^n_t:=V^n_{\nt}$
with
$V^n_k:=\psi_{n,k}(U^n_0,...,U^n_k)$, $\cV:=\Psi(\cU)$,
$\cY^n_t:=\sum_{k=1}^{\nt}V^n_{k-1}\otimes U^n_k$, and
$\cY_t:=\int_0^t\cV_s\otimes\dd\cU_s$. As an application of these
results,
first a Feller type diffusion approximation is derived for critical
multitype
branching processes with immigration if the offspring mean matrix is
primitive,
then the asymptotic behavior of the conditional least squares
estimator of the
offspring mean matrix is established.
http://front.math.ucdavis.edu/math.PR/0701803
---------------------------------------------------------------
5172. THE CUTOFF PHENOMENON FOR RANDOMIZED RIFFLE SHUFFLES
Guan-Yu Chen and Laurent Saloff-Coste
We study the cutoff phenomenon for generalized riffle shuffles where,
at each
step, the deck of cards is cut into a random number of packs of
multinomial
sizes which are then riffled together.
http://front.math.ucdavis.edu/math.PR/0701827
---------------------------------------------------------------
5173. M/M/$\INFTY$ QUEUES IN QUASI-MARKOVIAN RANDOM ENVIRONMENT
B. D'Auria
In this paper we investigate an M/M/$\infty$ queue whose parameters
depend on
an external random environment that we assume to be a quasi-Markovian
process
with finite state space. For this model we show a recursive formula
that allows
to compute all the factorial moments for the number of customers in
the system
in steady state. The used technique is based on the calculation of
the row
moments of the area of a bidimensional random set. Finally some
examples where
it is possible to get explicit formulas are given together with
comparisons
with previous known results.
http://front.math.ucdavis.edu/math.PR/0701842
---------------------------------------------------------------
5174. BSDES WITH STOCHASTIC LIPSCHITZ CONDITION AND QUADRATIC PDES IN
HILBERT SPACES
Philippe Briand (IRMAR) and Fulvia Confortola
This paper is devoted to the study of the differentiability of
solutions to
real-valued backward stochastic differential equations (BSDEs for
short) with
quadratic generators driven by a cylindrical Wiener process. The main
novelty
of this problem consists in the fact that the gradient equation of a
quadratic
BSDE has generators which satisfy stochastic Lipschitz conditions
involving BMO
martingales. We show some applications to the nonlinear Kolmogorov
equations.
http://front.math.ucdavis.edu/math.PR/0701849
---------------------------------------------------------------
5175. EXTREMAL PROBABILISTIC PROBLEMS AND HOTELLING'S T^2 TEST UNDER
SYMMETRY CONDITION
Iosif Pinelis
We consider Hotelling's T^2 statistic for an arbitrary d-dimensional
sample.
If the sampling is not too deterministic or inhomogeneous, then under
zero
means hypothesis, T^2 tends to \chi^2_d in distribution. We show that
a test
for the orthant symmetry condition introduced by Efron can be
constructed which
does not essentially differ from the one based on \chi^2_d and at the
same time
is applicable not only for large random homogeneous samples but for all
multidimensional samples without exceptions. The main assertions have
the form
of inequalities, not that of limit theorems; these inequalities are
exact
representing the solutions to certain extremal problems. Let us also
mention an
auxiliary result which itself may be of interest: \chi_d-(d-1)^{1/2}
decreases
in distribution in d to its limit N(0,1/2).
http://front.math.ucdavis.edu/math.ST/0701806
---------------------------------------------------------------
5176. RATE OF CONVERGENCE OF PENALIZED-LIKELIHOOD CONTEXT TREE
ESTIMATORS
Florencia G. Leonardi
We find upper bounds for the probability of error of the penalized-
likelihood
type context tree estimators, where the trees are not assumed to be
finite.
This estimators includes the well-known Bayesian Information
Criterion (BIC).
We show that the maximal decay for the probability of error can be
achieved
with a penalized term of the form $n^\alpha$, with $0 < \alpha < 1$.
http://front.math.ucdavis.edu/math.ST/0701810
---------------------------------------------------------------
5177. DETERMINISTIC MODAL BAYESIAN LOGIC: DERIVE THE BAYESIAN
INFERENCE WITHIN THE MODAL LOGIC T
Frederic Dambreville (DGA/CTA/DT/GIP)
In this paper a conditional logic is defined and studied. This
conditional
logic, DmBL, is constructed as a deterministic counterpart to the
Bayesian
conditional. The logic is unrestricted, so that any logical
operations are
allowed. A notion of logical independence is also defined within the
logic
itself. This logic is shown to be non-trivial and is not reduced to
classical
propositions. A model is constructed for the logic. Completeness
results are
proved. It is shown that any unconditioned probability can be
extended to the
whole logic DmBL. The Bayesian conditional is then recovered from the
probabilistic DmBL. At last, it is shown why DmBL is compliant with
Lewis'
triviality.
http://front.math.ucdavis.edu/math.LO/0701801
---------------------------------------------------------------
5178. FREE DIFFUSIONS AND MATRIX MODELS WITH STRICTLY CONVEX INTERACTION
A. Guionnet and D. Shlyakhtenko
We study solutions to the free stochastic differential equation $dX_t
= dS_t
- \half DV(X_t)dt$, where $V$ is a locally convex polynomial
potential in $m$
non-commuting variables. We show that for self-adjoint $V$, the law $
\mu_V$ of
a stationary solution is the limit law of a random matrix model, in
which an
$m$-tuple of self-adjoint matrices are chosen according to the law $
\exp(-N
\textrm{Tr}(V(A_1,...,A_m)))dA_1... dA_m$. We show that if $V=V_\beta
$ depends
on complex parameters $\beta_1,...,\beta_k$, then the law $\mu_V$ is
analytic
in $\beta$ at least for those $\beta$ for which $V_\beta$ is locally
convex. In
particular, this gives information on the region of convergence of the
generating function for planar maps.
We show that the solution $dX_t$ has nice convergence properties
with respect
to the operator norm. This allows us to derive several properties of
$C^*$ and
$W^*$ algebras generated by an $m$-tuple with law $\mu_V$. Among them
is lack
of projections, exactness, the Haagerup property, and embeddability
into the
ultrapower of the hyperfinite II$_1$ factor. We show that the
microstates free
entropy $\chi(\tau_V)$ is finite.
A corollary of these results is the fact that the support of the
law of any
self-adjoint polynomial in $X_1,...,X_n$ under the law $\mu_V$ is
connected,
vastly generalizing the case of a single random matrix.
http://front.math.ucdavis.edu/math.OA/0701787
---------------------------------------------------------------
5179. CLASSICAL AND VARIATIONAL DIFFERENTIABILITY OF BSDES WITH
QUADRATIC GROWTH
Stefan Ankirchner and Peter Imkeller and Goncalo Reis
We consider Backward Stochastic Differential Equations (BSDE) with
generators
that grow quadratically in the control variable. In a more abstract
setting, we
first allow both the terminal condition and the generator to depend
on a vector
parameter $x$. We give sufficient conditions for the solution pair of
the BSDE
to be differentiable in $x$. These results can be applied to systems of
forward-backward SDE. If the terminal condition of the BSDE is given
by a
sufficiently smooth function of the terminal value of a forward SDE,
then its
solution pair is differentiable with respect to the initial vector of
the
forward equation. Finally we prove sufficient conditions for
solutions of
quadratic BSDE to be differentiable in the variational sense (Malliavin
differentiable).
http://front.math.ucdavis.edu/math.PR/0701875
---------------------------------------------------------------
5180. PROPAGATION OF CHAOS AND POINCAR\'{E} INEQUALITIES FOR A SYSTEM
OF PARTICLES INTERACTING THROUGH THEIR CDF
Benjamin Jourdain (CERMICS) and Florent Malrieu (IRMAR)
In the particular case of a concave flux function, we are interested
in the
long time behaviour of the nonlinear process associated to the one-
dimensional
viscous scalar conservation law. We also consider the particle system
obtained
by remplacing the cumulative distribution function in the drift
coefficient of
this nonlinear process by the empirical cdf. We first obtain
trajectorial
propagation of chaos result. Then, Poincar\'{e} inequalities are used
to get
explicit estimates concerning the long time behaviour of both the
nonlinear
process and the particle system.
http://front.math.ucdavis.edu/math.PR/0701879
---------------------------------------------------------------
5181. LOCAL GAUSSIAN FLUCTUATIONS IN THE AIRY AND DISCRETE PNG PROCESSES
Jonas H\"agg
We prove that the Airy process, A(t), locally fluctuates like a Brownian
motion. In the same spirit we also show that in a certain scaling
limit, the so
called discrete polynuclear growth (PNG) process behaves like a Brownian
motion.
http://front.math.ucdavis.edu/math.PR/0701880
---------------------------------------------------------------
5182. RICCI CURVATURE OF MARKOV CHAINS ON METRIC SPACES
Yann Ollivier
We define the Ricci curvature of Markov chains on metric spaces as a
local
contraction coefficient of the random walk acting on the space of
probability
measures equipped with a Wasserstein transportation distance. For
Brownian
motion on a Riemannian manifold this gives back the value of Ricci
curvature of
a tangent vector. Examples of positively curved spaces for this
definition
include the discrete cube and discrete versions of the Ornstein--
Uhlenbeck
process. Moreover this generalization is consistent with the Bakry--
\'Emery
Ricci curvature for Brownian motion with a drift on a Riemannian
manifold.
Positive Ricci curvature is easily shown to imply a spectral gap
and a
L\'evy--Gromov-like Gaussian concentration theorem. These bounds are
sharp in
several interesting examples.
http://front.math.ucdavis.edu/math.PR/0701886
---------------------------------------------------------------
5183. MEASURE-PRESERVING TRANSFORMATIONS OF VOLTERRA GAUSSIAN
PROCESSES AND RELATED BRIDGES
Celine Jost
We consider Volterra Gaussian processes on [0,T], where T>0 is a
fixed time
horizon. These are processes of type X_t=\int^t_0 z_X(t,s)dW_s, t\in
[0,T],
where z_X is a square-integrable kernel, and W is a standard Brownian
motion.
An example is fractional Brownian motion. By using classical
techniques from
operator theory, we derive measure-preserving transformations of X,
and their
inherently related bridges of X. As a closely connected result, we
obtain a
Fourier-Laguerre series expansion for the first Wiener chaos of a
Gaussian
martingale over [0,\infty).
http://front.math.ucdavis.edu/math.PR/0701888
---------------------------------------------------------------
5184. EXPANSION PROPERTIES OF A RANDOM REGULAR GRAPH AFTER RANDOM
VERTEX DELETIONS
Catherine Greenhill (University of New South Wales) and Fred B.
Holt (University of Washington), Nicholas Wormald (University of
Waterloo)
We investigate the following vertex percolation process. Starting with a
random regular graph of constant degree, delete each vertex
independently with
probability p, where p=n^{-alpha} and alpha=alpha(n) is bounded away
from 0. We
show that a.a.s. the resulting graph has a connected component of
size n-o(n)
which is an expander, and all other components are trees of bounded
size.
Sharper results are obtained with extra conditions on alpha. These
results have
an application to the cost of repairing a certain peer-to-peer
network after
random failures of nodes.
http://front.math.ucdavis.edu/math.CO/0701863
---------------------------------------------------------------
5185. RECORD INDICES AND AGE-ORDERED FREQUENCIES IN GIBBS RANDOM
PARTITIONS
Robert C. Griffiths and Dario Span\'{o}
The distribution of age-ordered frequencies arising from an exchangeable
Gibbs partition is studied in relation with the distribution of the
positions
at which new mutations appear in a sample.
http://front.math.ucdavis.edu/math.PR/0701897
---------------------------------------------------------------
5186. DIFFERENTIATING SIGMA-FIELDS FOR GAUSSIAN AND SHIFTED GAUSSIAN
PROCESSES
S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA) and Giovanni
Peccati (LSTA)
We study the notions of differentiating and non-differentiating sigma-
fields
in the general framework of (possibly drifted) Gaussian processes, and
characterize their invariance properties under equivalent changes of
probability measure. As an application, we investigate the class of
stochastic
derivatives associated with shifted fractional Brownian motions. We
finally
establish conditions for the existence of a jointly measurable
version of the
differentiated process, and we outline a general framework for
stochastic
embedded equations.
http://front.math.ucdavis.edu/math.PR/0701910
---------------------------------------------------------------
5187. LOCAL LIMIT THEOREMS FOR LADDER EPOCHS
Vladimir Vatutin and Vitali Wachtel
Let {S_n, n=0,1,2,...} be a random walk generated by a sequence of
i.i.d.
random variables X_1, X_2,... and let tau be the first descending
ladder epoch.
Assuming that the distribution of X_1 belongs to the domain of
attraction of an
alpha-stable law, we study the asymptotic behavior of P(tau=n).
http://front.math.ucdavis.edu/math.PR/0701914
---------------------------------------------------------------
5188. PROLIFERATING PARASITES IN DIVIDING CELLS : KIMMEL'S BRANCHING
MODEL REVISITED
Vincent Bansaye (PMA)
We consider a branching model introduced by M. Kimmel for cell
division with
parasite infection. Cells contain proliferating parasites which are
shared
randomly between the two daughter cells when they divide. We
determine the
probability that the organism recovers, meaning that the asymptotic
proprotion
of contaminated cells vanishes. We study the tree of contaminated
cells, give
the asymptotic number of contaminated cells and the asymptotic
proportions of
contaminated cells with a given number of parasites. This depends on
domains
inherited from the behavior of branching processes in random
environment (BPRE)
and given by the bivariate value of the means of parasite offsprings.
In one of
these domains, the convergence of proportions holds in probability,
the limit
is deterministic and given by the Yaglom quasistationary
distribution. Moreover
we get an interpretation of the limit of the Q-process as the size-
biased
quasistationary distribution.
http://front.math.ucdavis.edu/math.PR/0701917
---------------------------------------------------------------
5189. ON LOWER LIMITS AND EQUIVALENCES FOR DISTRIBUTION TAILS OF
RANDOMLY STOPPED SUMS
Denis Denisov and Serguei Foss and Dmitry Korshunov
For a distribution $F^{*\tau}$ of a random sum $S_\tau=\xi_1+...+\xi_
\tau$ of
i.i.d. random variables with a common distribution $F$ on the half-line
$[0,\infty)$, we study the limits of the ratios of tails
$\bar{F^{*\tau}}(x)/\bar F(x)$ as $x\to\infty$ (here $\tau$ is an
independent
counting random variable). We also consider applications of obtained
results to
random walks, compound Poisson distributions, infinitely divisible
laws, and
sub-critical branching processes.
http://front.math.ucdavis.edu/math.PR/0701920
---------------------------------------------------------------
5190. ON SEVERAL TWO-BOUNDARY PROBLEMS FOR A PARTICULAR CLASS OF L
\'{E}VY PROCESSES
Tetyana Kadankova and No\"{e}l Veraverbeke
Several two-boundary problems are solved for a special L\'{e}vy
process: the
Poisson process with an exponential component. The jumps of this
process are
controlled by a homogeneous Poisson process, the positive jump size
distribution is arbitrary, while the distribution of the negative
jumps is
exponential. Closed form expressions are obtained for the integral
transforms
of the joint distribution of the first exit time from an interval and
the value
of the overshoot through boundaries at the first exit time. Also the
joint
distribution of the first entry time into the interval and the value
of the
process at this time instant are determined in terms of integral
transforms.
http://front.math.ucdavis.edu/math.PR/0701924
---------------------------------------------------------------
5191. KERNEL METHODS IN MACHINE LEARNING
Thomas Hofmann and Bernhard Sch\"olkopf and Alexander J. Smola
We review machine learning methods employing positive definite
kernels. These
methods formulate learning and estimation problems in a reproducing
kernel
Hilbert space (RKHS) of functions defined on the data domain,
expanded in terms
of a kernel. Working in linear spaces of function has the benefit of
facilitating the construction and analysis of learning algorithms
while at the
same time allowing large classes of functions. The latter include
nonlinear
functions as well as functions defined on non-vectorial data. We
cover a wide
range of methods, ranging from binary classifiers to sophisticated
methods for
estimation with structured data.
http://front.math.ucdavis.edu/math.ST/0701907
---------------------------------------------------------------
5192. EVALUATION OF FORMAL POSTERIOR DISTRIBUTIONS VIA MARKOV CHAIN
ARGUMENTS
Morris L. Eaton and James P. Hobert and Galin L. Jones and Wen-Lin Lai
We consider evaluation of proper posterior distributions obtained from
improper prior distributions. Our context is estimating a bounded
function
$\phi$ of a parameter when the loss is quadratic. If the posterior
mean of
$\phi$ is admissible for all bounded $\phi$ the posterior is \textit
{strongly
admissible}. In this paper, we present sufficient conditions for strong
admissibility. These conditions involve the recurrence of a symmetric
Markov
chain associated with the estimation problem. We develop general
sufficient
conditions for recurrence of general state space Markov chains that
are also of
independent interest. Our main example concerns the $p$-dimensional
multivariate normal distribution with mean vector $\theta$ when the
prior
distribution has the form $g_{0}(\theta) d\theta$ on the parameter space
$\mathbb{R}^{p}$. Conditions on $g_{0}$ for strong admissibility of the
posterior are provided.
http://front.math.ucdavis.edu/math.ST/0701938
---------------------------------------------------------------
5193. EXPONENTIAL CONTROL OF OVERLAP IN THE REPLICA METHOD FOR P-
SPIN SHERRINGTON-KIRKPATRICK MODEL
Dmitry Panchenko
Recently, Michel Talagrand computed the large deviations limit
$\lim_{N\to\infty}(Na)^{-1}\log \e Z_N^a$ for the moments of the
partition
function $Z_N$ in the Sherrington-Kirkpatrick model for all real $a
\geq 0.$ For
$a\geq 1$ the limit is given by Guerra's inverse bound and this
result extends
the classical physicist's replica method that corresponds to integer
$a.$ We
give a new proof for $a\geq 1$ in the case of the pure $p$-spin SK
model that
provides a strong exponential control of the overlap.
http://front.math.ucdavis.edu/math-ph/0701074
---------------------------------------------------------------
5194. ASYMPTOTICS OF NON-INTERSECTING BROWNIAN MOTIONS AND A 4 X 4
RIEMANN-HILBERT PROBLEM
Evi Daems and Arno Kuijlaars and and Wim Veys
We consider n one-dimensional Brownian motions, such that n/2 Brownian
motions start at time t=0 in the starting point a and end at time t=1
in the
endpoint b and the other n/2 Brownian motions start at time t=0 at
the point -a
and end at time t=1 in the point -b, conditioned that the n Brownian
paths do
not intersect in the whole time interval (0,1). The correlation
functions of
the positions of the non-intersecting Brownian motions have a
determinantal
form with a kernel that is expressed in terms of multiple Hermite
polynomials
of mixed type. We analyze this kernel in the large n limit for the
case ab<1/2.
We find that the limiting mean density of the positions of the
Brownian motions
is supported on one or two intervals and that the correlation kernel
has the
usual scaling limits from random matrix theory, namely the sine
kernel in the
bulk and the Airy kernel near the edges.
http://front.math.ucdavis.edu/math.CV/0701923
---------------------------------------------------------------
5195. A COMBINATORIAL METHOD FOR CALCULATING THE MOMENTS OF L\'EVY AREA
Daniel Levin and Mark Wildon
We present a new way to compute the moments of the L\'evy area of a
two-dimensional Brownian motion. Our approach uses iterated integrals
and
combinatorial arguments involving the shuffle product.
http://front.math.ucdavis.edu/math.PR/0702002
---------------------------------------------------------------
5196. FINITE SIZE SCALING FOR THE CORE OF LARGE RANDOM HYPERGRAPHS
Amir Dembo and Andrea Montanari
The (two) core of an hyper-graph is the maximal collection of hyper-
edges
within which no vertex appears only once. It is of importance in
tasks such as
efficiently solving a large linear system over GF[2], or iterative
decoding of
low-density parity-check codes used over the binary erasure channel.
Similar
structures emerge in a variety of NP-hard combinatorial optimization and
decision problems, from vertex cover to satisfiability.
For a uniformly chosen random hyper-graph of m=n\rho vertices and n
hyper-edges, each consisting of the same fixed number l >= 3 of
vertices, the
size of the core exhibits for large n a first order phase transition,
changing
from o(n) for rho> rho_c to a positive fraction of n for rho<rho_c,
with a
transition window size Theta(n^{-1/2}) around rho_c>0. Analyzing the
corresponding `leaf removal' algorithm, we determine the associated
finite size
scaling behavior. In particular, if rho is inside the scaling window
(more
precisely, rho = rho_c+r n^{-1/2}, the probability of having a core
of size
Theta(n) has a limit strictly between 0 and 1, and a leading
correction of
order Theta(n^{-1/6}). The correction admits a sharp characterization
in terms
of the distribution of a Brownian motion with quadratic shift, from
which it
inherits the scaling with n. This behavior is expected to be
universal for wide
collection of combinatorial problems.
http://front.math.ucdavis.edu/math.PR/0702007
---------------------------------------------------------------
5197. ON STEIN'S METHOD AND PERTURBATIONS
Andrew D. Barbour and Vydas Cekanavicius and Aihua Xia
Stein's (1972) method is a very general tool for assessing the
quality of
approximation of the distribution of a random element by another, often
simpler, distribution. In applications of Stein's method, one needs to
establish a Stein identity for the approximating distribution, solve
the Stein
equation and estimate the behaviour of the solutions in terms of the
metrics
under study. For some Stein equations, solutions with good properties
are
known; for others, this is not the case. Barbour and Xia (1999)
introduced a
perturbation method for Poisson approximation, in which Stein
identities for a
large class of compound Poisson and translated Poisson distributions
are viewed
as perturbations of a Poisson distribution. In this paper, it is
shown that the
method can be extended to very general settings, including
perturbations of
normal, Poisson, compound Poisson, binomial and Poisson process
approximations
in terms of various metrics such as the Kolmogorov, Wasserstein and
total
variation metrics. Examples are provided to illustrate how the general
perturbation method can be applied.
http://front.math.ucdavis.edu/math.PR/0702008
---------------------------------------------------------------
5198. ON A RANDOMIZED PNG MODEL WITH A COLUMNAR DEFECT
Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (BR-IMPA) and
Maria Eulalia Vares (BR-CBPF)
We study a variant of poly-nuclear growth where the level boundaries
perform
continuous-time, discrete-space random walks, and study how its
asymptotic
behavior is affected by the presence of a columnar defect on the
line. We prove
that there is a non-trivial phase transition in the strength of the
perturbation, above which the law of large numbers for the height
function is
modified.
http://front.math.ucdavis.edu/math.PR/0702012
---------------------------------------------------------------
5199. A FUNCTIONAL CLT FOR THE OCCUPATION TIME OF STATE-DEPENDENT
BRANCHING RANDOM WALK
Matthias Birkner and Iljana Z\"ahle
We show that the centred occupation time process of the origin of a
system of
critical binary branching random walks in dimension $d \ge 3$,
started off
either from a Poisson field or in equilibrium, when suitably normalised,
converges to a Brownian motion in $d \ge 4$. In $d=3$, the limit
process is
fractional Brownian motion with Hurst parameter 3/4 when starting in
equilibrium, and a related Gaussian process when starting from a
Poisson field.
For (dependent) branching random walks with state dependent branching
rate we
obtain convergence in f.d.d. to the same limit process, and for $d=3$
also a
functional limit theorem.
http://front.math.ucdavis.edu/math.PR/0702020
---------------------------------------------------------------
5200. GRAPHS WITH SPECIFIED DEGREE DISTRIBUTIONS, SIMPLE EPIDEMICS
AND LOCAL VACCINATION STRATEGIES
Tom Britton and Svante Janson and Anders Martin-Lof
Consider a random graph, having a pre-specified degree distribution F
but
other than that being uniformly distributed, describing the social
structure
(friendship) in a large community. Suppose one individual in the
community is
externally infected by an infectious disease and that the disease has
its
course by assuming that infected individuals infect their not yet
infected
friends independently with probability p. For this situation the paper
determines R_0 and tau_0, the basic reproduction number and the
asymptotic
final size in case of a major outbreak. Further, the paper looks at some
different local vaccination strategies where individuals are chosen
randomly
and vaccinated, or friends of the selected individuals are
vaccinated, prior to
the introduction of the disease. For the studied vaccination
strategies the
paper determines R_v: the reproduction number, and tau_v: the
asymptotic final
proportion infected in case of a major outbreak, after vaccinating a
fraction
v.
http://front.math.ucdavis.edu/math.PR/0702021
---------------------------------------------------------------
5201. WIGNER RANDOM MATRICES WITH NON-SYMMETRICALLY DISTRIBUTED ENTRIES
Sandrine Peche and Alexander Soshnikov
We show that the spectral radius of an $N\times N$ random symmetric
matrix
with i.i.d. bounded centered but non-symmetrically distributed
entries is
bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon}), $ where $
\sigma^2 $
is the variance of the matrix entries and $\epsilon $ is an arbitrary
small
positive number. Our bound improves the earlier results by Z.F\"{u}
redi and
J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).
http://front.math.ucdavis.edu/math.PR/0702035
---------------------------------------------------------------
5202. ON THE VARIANCE OF THE OPTIMAL ALIGNMENT SCORE FOR AN
ASYMMETRIC SCORING FUNCTION
Christian Houdr\'e and Heinrich Matzinger
We investigate the variance of the optimal alignment score of two
independent
iid binary, with parameter 1/2, sequences of length $n$. The scoring
function
is such that one letter has a somewhat larger score than the other
letter. In
this setting, we prove that the variance is of order $n$, and this
confirms
Waterman's conjecture in this case.
http://front.math.ucdavis.edu/math.PR/0702036
---------------------------------------------------------------
5203. A LARGE DEVIATION PRINCIPLE IN H\"OLDER NORM FOR MULTIPLE
FRACTIONAL INTEGRALS
Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
For a fractional Brownian motion $B^H$ with Hurst parameter
$H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a
simplex
are constructed and the regularity of their sample paths are studied.
Then, it
is proved that the family of probability laws of the processes
obtained by
replacing $B^H$ by $\epsilon^{{1/2}} B^H$ satisfies a large deviation
principle
in H\"older norm. The definition of the multiple integrals relies upon a
representation of the fractional Brownian motion in terms of a
stochastic
integral with respect to a standard Brownian motion. For the large
deviation
principle, the abstract general setting given by Ledoux in [Lecture
Notes in
Math., vol. 1426 (1990) 1-14] is used.
http://front.math.ucdavis.edu/math.PR/0702049
---------------------------------------------------------------
5204. H\"{O}LDER REGULARITY FOR OPERATOR SCALING STABLE RANDOM FIELDS
Hermine Bierm\'{e} (MAP5) and C\'{e}line Lacaux (IECN)
We investigate the sample paths regularity of operator scaling alpha-
stable
random fields. Such fields were introduced as anisotropic
generalizations of
self-similar fields and satisfy a scaling property for a real matrix
E. In the
case of harmonizable operator scaling random fields, the sample paths
are
locally H\"{o}lderian and their H\"{o}lder regularity is
characterized by the
eigen decomposition with respect to E. In particular, the directional
H\"{o}lder regularity may vary and is given by the eigenvalues of E.
In the
case of moving average operator scaling random alpha-stable random
fields, with
0<alpha<2, the sample paths are almost surely discontinous.
http://front.math.ucdavis.edu/math.PR/0702050
---------------------------------------------------------------
5205. LARGE DEVIATIONS FOR EMPIRICAL PATH MEASURES IN CYCLES OF
INTEGER PARTITIONS
Stefan Adams
Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on
some
fixed time interval $[0,\beta]$ with symmetrised initial-terminal
condition.
That is, for any $i$, the terminal location of the $i$-th motion is
affixed to
the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a
uniformly
distributed random permutation of $1,...,N$.
In this paper, we describe the large-N behaviour of the empirical
path
measure (the mean of the Dirac measures in the $N$ paths) when $
\Lambda\uparrow\mathbb{R}^d $ and $ N/|\Lambda|\to\rho $. The rate
function is
given as a variational formula involving a certain entropy functional
and a
Fenchel-Legendre transform.
Depending on the dimension and the density $ \rho $, there is phase
transition behaviour for the empirical path measure. For certain
parameters
(high density, large time horizon) and dimensions $ d\ge 3 $ the
empirical path
measure is not supported on all paths $ [0,\infty)\to\mathbb{R}^d $
which
contain a bridge path of any finite multiple of the time horizon $ [0,
\beta] $.
For dimensions $ d=1,2 $, and for small densities and small time
horizon $
[0,\beta] $ in dimensions $ d\ge 3$, the empirical path measure is
supported on
those paths. In the first regime a finite fraction of the motions
lives in
cycles of infinite length.
We outline that this transition leads to an empirical path measure
interpretation of {\it Bose-Einstein condensation}, known for systems of
Bosons.
http://front.math.ucdavis.edu/math.PR/0702053
---------------------------------------------------------------
5206. MIXTURES IN NON STABLE LEVY PROCESSES
Nicola Cufaro Petroni
We analyze the Levy processes produced by means of two interconnected
classes
of non stable, infinitely divisible distribution: the Variance Gamma
and the
Student laws. While the Variance Gamma family is closed under
convolution, the
Student one is not: this makes its time evolution more complicated.
We prove
that -- at least for one particular type of Student processes
suggested by
recent empirical results, and for integral times -- the distribution
of the
process is a mixture of other types of Student distributions,
randomized by
means of a new probability distribution. The mixture is such that
along the
time the asymptotic behavior of the probability density functions always
coincide with that of the generating Student law. We put forward the
conjecture
that this can be a general feature of the Student processes. We
finally analyze
the Ornstein--Uhlenbeck process driven by our Levy noises and show a few
simulation of it.
http://front.math.ucdavis.edu/math.PR/0702058
---------------------------------------------------------------
5207. LINE-OF-SIGHT PERCOLATION
Bela Bollobas and Svante Janson and Oliver Riordan
Given $\omega\ge 1$, let $Z^2_{(\omega)}$ be the graph with vertex
set $Z^2$
in which two vertices are joined if they agree in one coordinate and
differ by
at most $\omega$ in the other. (Thus $Z^2_{(1)}$ is precisely $Z^2$.)
Let
$p_c(\omega)$ be the critical probability for site percolation in
$Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi
and
Debany, we show that $\lim_{\omega\to\infty} \omega\pc(\omega)=\log
(3/2)$. We
also prove analogues of this result on the $n$-by-$n$ grid and in higher
dimensions, the latter involving interesting connections to Gilbert's
continuum
percolation model. To prove our results, we explore the component of
the origin
in a certain non-standard way, and show that this exploration is well
approximated by a certain branching process.
http://front.math.ucdavis.edu/math.PR/0702061
---------------------------------------------------------------
5208. ANCESTRAL PROCESSES WITH SELECTION: BRANCHING AND MORAN MODELS
E. Baake and R. Bialowons
We consider two versions of stochastic population models with
mutation and
selection. The first approach relies on a multitype branching
process; here,
individuals reproduce and change type (i.e., mutate) independently of
each
other, without restriction on population size. We analyze the
equilibrium
behaviour of this model, both in the forward and in the backward
direction of
time; the backward point of view emerges if the ancestry of
individuals chosen
randomly from the present population is traced back into the past.
The second approach is the Moran model with selection. Here, the
population
has constant size N. Individuals reproduce (at rates depending on
their types),
the offspring inherits the parent's type, and replaces a randomly chosen
individual (to keep population size constant). Independently of the
reproduction process, individuals can change type. As in the
branching model,
we consider the ancestral lines of single individuals chosen from the
equilibrium population. We use analytical results of Fearnhead (2002) to
determine the explicit properties, and parameter dependence, of the
ancestral
distribution of types, and its relationship with the stationary
distribution in
forward time.
http://front.math.ucdavis.edu/q-bio.PE/0702002
---------------------------------------------------------------
5209. ISOSPIN ASYMMETRY IN NUCLEI AND NUCLEAR SYMMETRY ENERGY
Tapan Mukhopadhyay and D.N. Basu
Binding energy of isospin asymmetric nuclei can be accessed with
minimally
modified formula along the lines of the liquid droplet model by
partitioning
the symmetry term into volume and surface terms. The volume symmetry
energy
coefficient extracted from finite nuclei provides a constraint on the
nuclear
symmetry energy. This approach also yields the neutron skin of a
finite nucleus
through its relationship with the volume and surface symmetry terms
and the
Coulomb energy coefficient. The symmetry energy at saturation density
obtained
from the isoscalar as well as isovector components of the density
dependent M3Y
effective interaction is found to be in close agreement with the volume
symmetry energy coefficient extracted from the measured atomic masses.
http://front.math.ucdavis.edu/nucl-th/0605001
---------------------------------------------------------------
5210. COMPUTING THE LOEWNER DRIVING PROCESS OF RANDOM CURVES IN THE
HALF PLANE
Tom Kennedy
We simulate several models of random curves in the half plane and
numerically
compute their stochastic driving process (as given by the Loewner
equation).
Our models include models whose scaling limit is the Schramm-Loewner
evolution
(SLE) and models for which it is not. We study several tests of
whether the
driving process is Brownian motion. We find that just testing the
normality of
the process at a fixed time is not effective at determining if the
process is
Brownian motion. Tests that involve the independence of the
increments of
Brownian motion are much more effective. We also study the zipper
algorithm for
numerically computing the driving function of a simple curve. We give an
implementation of this algorithm which runs in a time O(N^1.35)
rather than the
usual O(N^2), where N is the number of points on the curve.
http://front.math.ucdavis.edu/math.PR/0702071
---------------------------------------------------------------
5211. LIMIT THEOREMS ON LOCALLY COMPACT ABELIAN GROUPS
Matyas Barczy and Alexander Bendikov and Gyula Pap
We prove limit theorems for row sums of a rowwise independent
infinitesimal
array of random variables with values in a locally compact Abelian
group. First
we give a proof of Gaiser's theorem, since it does not have an easy
access and
it is not complete. This theorem gives sufficient conditions for
convergence of
the row sums, but the limit measure can not have a nondegenerate
idempotent
factor. Then we prove necessary and sufficient conditions for
convergence of
the row sums, where the limit measure can be also a nondegenerate
Haar measure
on a compact subgroup. Finally, we investigate special cases: the
torus group,
the group of p-adic integers and the p-adic solenoid.
http://front.math.ucdavis.edu/math.PR/0702078
---------------------------------------------------------------
5212. ON A NON-CLASSICAL INVARIANCE PRINCIPLE
Youri Davydov (Universite de Lille 1) and Vladimir Rotar (San Diego
State University)
We consider the invariance principle without the classical condition of
asymptotic negligibility of individual terms. More precisely, we
explore the
difference of the following two distributions in the space C (of
continuous
functions on [0,1]). The first is the distribution of the continuous
piecewise
linear partial-sum process generated by a sequence of independent random
variables, and the second is the distribution of the similar process
generated
by the sequence of normal r.v.'s with the same first two moments. The
novelty
is that the condition of negligibility of the r.v.'s is not imposed. We
establish a necessary and sufficient condition of the weak
convergence of the
difference mentioned to zero measure in C.
http://front.math.ucdavis.edu/math.PR/0702085
---------------------------------------------------------------
5213. DIVERGENCE THEOREMS IN PATH SPACE III: HYPOELLIPTIC DIFFUSIONS
AND BEYOND
Denis Bell
Let $x$ denote a diffusion process defined on a closed compact
manifold. In
an earlier article, the author introduced a new approach to constructing
admissible vector fields on the associated space of paths, under the
assumption
of ellipticity of $x$. In this article, this method is extended to yield
similar results for degenerate diffusion processes. In particular, these
results apply to non-elliptic diffusions satisfying H\"ormander's
condition.
http://front.math.ucdavis.edu/math.PR/0702092
---------------------------------------------------------------
5214. EXTINCTION VERSUS UNBOUNDED GROWTH; HABILITATION THESIS OF THE
UNIVERSITY ERLANGEN-N\"URNBERG
Jan M. Swart
Certain Markov processes, or deterministic evolution equations, have the
property that they are dual to a stochastic process that exhibits
extinction
versus unbounded growth, i.e., the total mass in such a process
either becomes
zero, or grows without bounds as time tends to infinity. If this is
the case,
then this phenomenon can often be used to determine the invariant
measures, or
fixed points, of the process originally under consideration, and to
study
convergence to equilibrium. This principle, which has been known
since early
work on multitype branching processes, is here demonstrated on three new
examples with applications in the theory of interacting particle
systems.
http://front.math.ucdavis.edu/math.PR/0702095
---------------------------------------------------------------
5215. A NOTE ON ERGODIC TRANSFORMATIONS OF SELF-SIMILAR VOLTERRA
GAUSSIAN PROCESSES
Celine Jost
We derive a class of ergodic transformations of self-similar Gaussian
processes that are Volterra, i.e. of type X_t = int^t_0 z_X(t,s)dW_s,
t>0,
where z_X is a deterministic kernel and W is a standard Brownian motion.
http://front.math.ucdavis.edu/math.PR/0702096
---------------------------------------------------------------
5216. RANDOM WALK IN MARKOVIAN ENVIROMENT
Dmitry Dolgopyat and Gerhard Keller and and Carlangelo Liverani
We prove a quenched central limit theorem for random walks with bounded
increments in a randomly evolving environment on Zd. We assume that the
transition probabilities of the walk depend not too strongly on the
environment
and that the evolution of the environment is Markovian with strong
spatial and
temporal mixing properties.
http://front.math.ucdavis.edu/math.PR/0702100
---------------------------------------------------------------
5217. TAIL PROBABILITIES FOR INFINITE SERIES OF REGULARLY VARYING
RANDOM VECTORS
Henrik Hult and Gennady Samorodnitsky
A random vector $X$ with representation $X = \sum_{j \geq 0} A_j Z_j$ is
considered. Here $(Z_j)$ is a sequence of independent and identically
distributed random vectors and $(A_j)$ is a sequence of random matrices,
``predictable'' with respect to the sequence $(Z_j)$. The
distribution of $Z_1$
is assumed to be multivariate regular varying. Moment conditions on the
matrices $(A_j)$ are determined under which the distribution of $X$ is
regularly varying and, in fact, ``inherits'' its regular variation
from that of
$(Z_j)$'s. We compute the associated limiting measure. Examples
include linear
processes, random coefficient linear processes such as stochastic
recurrence
equations, random sums, and stochastic integrals.
http://front.math.ucdavis.edu/math.PR/0702112
---------------------------------------------------------------
5218. DYNAMICAL PROPERTIES OF A TAGGED PARTICLE IN THE TOTALLY
ASYMMETRIC SIMPLE EXCLUSION PROCESS WITH THE STEP-TYPE INITIAL
CONDITION
T. Imamura and T. Sasamoto
The one-dimensional totally asymmetric simple exclusion process
(TASEP) is
considered. We study the time evolution property of a tagged particle
in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the
dynamics of
TASEP to the Schur process, we show that the function is represented
as the
Fredholm determinant. We also study the scaling limit. The
universality of the
largest eigenvalue in the random matrix theory is realized in the
limit. When
the hopping rates of all particles are the same, it is found that the
joint
distribution function converges to that of the Airy process after the
time at
whichthe particle begins to move. On the other hand, when there are
several
particles with small hopping rate in front of a tagged particle, the
limiting
process changes at a certain time from the Airy process to the
process of the
largest eigenvalue in the Hermitian multi-matrix model with external
sources.
http://front.math.ucdavis.edu/math-ph/0702009
---------------------------------------------------------------
5219. MELLIN TRANSFORM AND SUBORDINATION LAWS IN FRACTIONAL
DIFFUSION PROCESSES
Francesco Mainardi and Gianni Pagnini and Rudolf Gorenflo
The Mellin transform is usually applied in probability theory to the
product
of independent random variables. In recent times the machinery of the
Mellin
transform has been adopted to describe the L\'evy stable
distributions, and
more generally the probability distributions governed by generalized
diffusion
equations of fractional order in space and/or in time. In these cases
the
related stochastic processes are self-similar and are simply referred
to as
fractional diffusion processes. We provide some integral formulas
involving the
distributions of these processes that can be interpreted in terms of
subordination laws.
http://front.math.ucdavis.edu/math.PR/0702133
---------------------------------------------------------------
5220. LOCAL ENERGY STATISTICS IN DIRECTED POLYMERS
Irina Kourkova (PMA)
Recently, Bauke and Mertens conjectured that the local statistics of
energies
in random spin systems with discrete spin space should, in most
circumstances,
be the same as in the random energy model. We show that this
conjecture holds
true as well for directed polymers in random environment. We also
show that,
under certain conditions, this conjecture holds for directed polymers
even if
energy levels that grow moderately with the volume of the system are
considered.
http://front.math.ucdavis.edu/math.PR/0702149
---------------------------------------------------------------
5221. STATISTICAL ANALYSIS OF THE DIFFIE-HELLMAN KEY EXCHANGE
PROTOCOL IN A FINITE GROUP
I. Florescu and A. Myasnikov and A. Mahalanobis
This paper presents a novel methodology to test the security of the
Diffie-Hellman public key exchange protocol. The security of many
cryptographic
schemes rely on the hardness of this problem. We are presenting a purely
statistical test to compare this problem in different groups. We are
using
groups included in the Zp group with p prime as a major example,
however the
methods presented are not restricted to these groups. The
presentation of the
results is primarily intended to introduce novel applications of
statistical
methodologies to the area of mathematical cryptography. As such we will
emphasize the cryptographical aspects of the work more than the
statistical
notions.
http://front.math.ucdavis.edu/math.ST/0702155
---------------------------------------------------------------
5222. PATHWISE INEQUALITIES FOR LOCAL TIME: APPLICATIONS TO
SKOROKHOD EMBEDDINGS AND OPTIMAL STOPPING
A.M.G.Cox and D.Hobson and J.Ob\l\'oj
We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+
F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero
and $M_t$
a local martingale. The concrete nature of the representation makes the
inequality useful for a variety of applications. In this work, we use
the
inequalities to derive constructions and optimality results of Vallois'
Skorokhod embeddings. We discuss their financial interpretation in
the context
of robust pricing and hedging of options written on the local time.
In the
final part of the paper we use the inequalities to solve a class of
optimal
stopping problems of the form $\sup_\tau E[F(L_\tau)-\int_0^\tau
\beta(B_s)ds]$. The solution is given via a minimal solution to a
system of
differential equations and thus resembles the maximality principle
described by
Peskir. Throughout, the emphasis is placed on the novelty and
simplicity of the
techniques.
http://front.math.ucdavis.edu/math.PR/0702173
---------------------------------------------------------------
5223. DIFFERENCE APPROXIMATION FOR LOCAL TIMES OF MULTIDIMENSIONAL
DIFFUSIONS
Alexey M. Kulik
We consider sequences of additive functionals of difference
approximations
for uniformly non-degenerate multidimensional diffusions. The
conditions are
given, sufficient for such a sequence to converge weakly to a W-
functional of
the limiting process. The class of the W-functionals, that can be
obtained as
the limiting ones, is completely described in the terms of the
associated
W-measures, and coincides with the class of the functionals that are
regular
w.r.t. the phase variable.
http://front.math.ucdavis.edu/math.PR/0702175
---------------------------------------------------------------
5224. DIFFUSION APPROXIMATION FOR EQUILIBRIUM KAWASAKI DYNAMICS IN
CONTINUUM
Y.G. Kondratiev and O.V. Kutoviy and E.W. Lytvynov
A Kawasaki dynamics in continuum is a dynamics of an infinite system of
interacting particles in $\mathbb R^d$ which randomly hop over the
space. In
this paper, we deal with an equilibrium Kawasaki dynamics which has a
Gibbs
measure $\mu$ as invariant measure. We study a diffusive limit of such a
dynamics, derived through a scaling of both the jump rate and time.
Under weak
assumptions on the potential of pair interaction, $\phi$, (in
particular,
admitting a singularity of $\phi$ at zero), we prove that, on a set
of smooth
local functions, the generator of the scaled dynamics converges to the
generator of an equilibrium diffusive dynamics of an infinite system of
interacting particles. If the set on which the generators converge is
a core
for the diffusion generator, the latter result implies the weak
convergence of
finite-dimensional distributions of the corresponding equilibrium
processes. In
particular, if the potential $\phi$ is from $C_{\mathrm b}^3(\R^d)$ and
sufficiently quickly converges to zero at infinity, we conclude from
a result
in [Choi {\it et al.}, {J. Math. Phys.} {39} (1998) 6509--6536] that the
convergence of processes holds when the limiting diffusion is the
gradient
stochastic dynamics.
http://front.math.ucdavis.edu/math.PR/0702178
---------------------------------------------------------------
5225. FINITE-SIZE EFFECTS FOR ANISOTROPIC BOOTSTRAP PERCOLATION:
LOGARITHMIC CORRECTIONS
Aernout C.D. van Enter and Tim Hulshof
In this note we analyze an anisotropic, two-dimensional bootstrap
percolation
model introduced by Gravner and Griffeath. We present upper and lower
bounds on
the finite-size effects. We discuss the similarities with the semi-
oriented
model introduced by Duarte.
http://front.math.ucdavis.edu/cond-mat/0702145
---------------------------------------------------------------
5226. PARABOLIC HARNACK INEQUALITY AND HEAT KERNEL ESTIMATES FOR
RANDOM WALKS WITH LONG RANGE JUMPS
M.T. Barlow and R.F. Bass and and T. Kumagai
We investigate the relationships between the parabolic Harnack
inequality,
heat kernel estimates, some geometric conditions, and some analytic
conditions
for random walks with long range jumps. Unlike the case of diffusion
processes,
the parabolic Harnack inequality does not, in general, imply the
corresponding
heat kernel estimates.
http://front.math.ucdavis.edu/math.PR/0702221
---------------------------------------------------------------
5227. RIGOROUS CONFIDENCE INTERVALS FOR CRITICAL PROBABILITIES
Oliver Riordan and Mark Walters
We use the method of Balister, Bollobas and Walters to give rigorous
99.9999%
confidence intervals for the critical probabilities for site and bond
percolation on the 11 Archimedean lattices. In our computer
calculations, the
emphasis is on simplicity and ease of verification, rather than
obtaining the
best possible results. Nevertheless, we obtain intervals of width at
most
0.0005 in all cases.
http://front.math.ucdavis.edu/math.PR/0702232
---------------------------------------------------------------
5228. A RANDOMIZED KACZMARZ ALGORITHM WITH EXPONENTIAL CONVERGENCE
Thomas Strohmer and Roman Vershynin
The Kaczmarz method for solving linear systems of equations is an
iterative
algorithm that has found many applications ranging from computer
tomography to
digital signal processing. Despite the popularity of this method, useful
theoretical estimates for its rate of convergence are still scarce. We
introduce a randomized version of the Kaczmarz method for consistent,
overdetermined linear systems and we prove that it converges with
expected
exponential rate. Furthermore, this is the first solver whose rate
does not
depend on the number of equations in the system. The solver does not
even need
to know the whole system, but only a small random part of it. It thus
outperforms all previously known methods on general extremely
overdetermined
systems. Even for moderately overdetermined systems, numerical
simulations as
well as theoretical analysis reveal that our algorithm can converge
faster than
the celebrated conjugate gradient algorithm. Furthermore, our theory and
numerical simulations confirm a prediction of Feichtinger et al. in
the context
of reconstructing bandlimited functions from nonuniform sampling.
http://front.math.ucdavis.edu/math.NA/0702226
---------------------------------------------------------------
5229. A SIMPLE PROOF OF KAIJSER'S UNIQUE ERGODICITY RESULT FOR HIDDEN
MARKOV $\ALPHA$-CHAINS
Fred Kochman and Jim Reeds
According to a 1975 result of T. Kaijser, if some nonvanishing
product of
hidden Markov model (HMM) stepping matrices is subrectangular, and the
underlying chain is aperiodic, the corresponding $\alpha$-chain has a
unique
invariant limiting measure $\lambda$. Here the $\alpha$-chain
$\{\alpha_n\}=\{(\alpha_{ni})\}$ is given by \[\alpha_{ni}=P(X_n=i|
Y_n,Y_{n-1},...),\] where $\{(X_n,Y_n)\}$ is a finite state HMM with
unobserved
Markov chain component $\{X_n\}$ and observed output component $\{Y_n
\}$. This
defines $\{\alpha_n\}$ as a stochastic process taking values in the
probability
simplex. It is not hard to see that $\{\alpha_n\}$ is itself a Markov
chain.
The stepping matrices $M(y)=(M(y)_{ij})$ give the probability that
$(X_n,Y_n)=(j,y)$, conditional on $X_{n-1}=i$. A matrix is said to be
subrectangular if the locations of its nonzero entries forms a cartesian
product of a set of row indices and a set of column indices.
Kaijser's result
is based on an application of the Furstenberg--Kesten theory to the
random
matrix products $M(Y_1)M(Y_2)... M(Y_n)$. In this paper we prove a
slightly
stronger form of Kaijser's theorem with a simpler argument,
exploiting the
theory of e chains.
http://front.math.ucdavis.edu/math.PR/0702248
---------------------------------------------------------------
5230. CONTINUOUS-TIME MEAN-VARIANCE EFFICIENCY: THE 80% RULE
Xun Li and Xun Yu Zhou
This paper studies a continuous-time market where an agent, having
specified
an investment horizon and a targeted terminal mean return, seeks to
minimize
the variance of the return. The optimal portfolio of such a problem
is called
mean-variance efficient \`{a} la Markowitz. It is shown that, when
the market
coefficients are deterministic functions of time, a mean-variance
efficient
portfolio realizes the (discounted) targeted return on or before the
terminal
date with a probability greater than 0.8072. This number is universal
irrespective of the market parameters, the targeted return and the
length of
the investment horizon.
http://front.math.ucdavis.edu/math.PR/0702249
---------------------------------------------------------------
5231. PERIODICITY IN THE TRANSIENT REGIME OF EXHAUSTIVE POLLING SYSTEMS
I. M. MacPhee and M. V. Menshikov and S. Popov and S. Volkov
We consider an exhaustive polling system with three nodes in its
transient
regime under a switching rule of generalized greedy type. We show
that, for the
system with Poisson arrivals and service times with finite second
moment, the
sequence of nodes visited by the server is eventually periodic almost
surely.
To do this, we construct a dynamical system, the triangle process,
which we
show has eventually periodic trajectories for almost all sets of
parameters and
in this case we show that the stochastic trajectories follow the
deterministic
ones a.s. We also show there are infinitely many sets of parameters
where the
triangle process has aperiodic trajectories and in such cases
trajectories of
the stochastic model are aperiodic with positive probability.
http://front.math.ucdavis.edu/math.PR/0702252
---------------------------------------------------------------
5232. SAMPLE PATH LARGE DEVIATIONS FOR MULTICLASS FEEDFORWARD
QUEUEING NETWORKS IN CRITICAL LOADING
Kurt Majewski
We consider multiclass feedforward queueing networks with first in
first out
and priority service disciplines at the nodes, and class dependent
deterministic routing between nodes. The random behavior of the
network is
constructed from cumulative arrival and service time processes which are
assumed to satisfy an appropriate sample path large deviation
principle. We
establish logarithmic asymptotics of large deviations for waiting
time, idle
time, queue length, departure and sojourn-time processes in critical
loading.
This transfers similar results from Puhalskii about single class
queueing
networks with feedback to multiclass feedforward queueing networks, and
complements diffusion approximation results from Peterson. An example
with
renewal inter arrival and service time processes yields the rate
function of a
reflected Brownian motion. The model directly captures stationary
situations.
http://front.math.ucdavis.edu/math.PR/0702256
---------------------------------------------------------------
5233. ON THE HAUSDORFF DIMENSION OF REGULAR POINTS OF INVISCID
BURGERS EQUATION WITH STABLE INITIAL DATA
Thomas Simon (DP)
Consider an inviscid Burgers equation whose initial data is a Levy a-
stable
process Z with a > 1. We show that when Z has positive jumps, the
Hausdorff
dimension of the set of Lagrangian regular points associated with the
equation
is strictly smaller than 1/a, as soon as a is close to 1. This gives
a negative
answer to a conjecture of Janicki and Woyczynski. Along the way, we
contradict
a recent conjecture of Z. Shi about the lower tails of integrated stable
processes.
http://front.math.ucdavis.edu/math.PR/0702260
---------------------------------------------------------------
5234. STOCHASTIC MODELS FOR PHYLOGENETIC TREES ON HIGHER-ORDER TAXA
David Aldous and Maxim Krikun and and Lea Popovic
Simple stochastic models for phylogenetic trees on species have been
well
studied. But much paleontology data concerns time series or trees on
higher-order taxa, and any broad picture of relationships between
extant groups
requires use of higher-order taxa. A coherent model for trees on
(say) genera
should involve both a species-level model and a model for the
classification
scheme by which species are assigned to genera. We present a general
framework
for such models, and describe three alternate classification schemes.
Combining
with the species-level model of Aldous-Popovic (2005), one gets
models for
higher-order trees, and we initiate analytic study of such models. In
particular we derive formulas for the lifetime of genera, for the
distribution
of number of species per genus, and for the offspring structure of
the tree on
genera.
http://front.math.ucdavis.edu/q-bio.PE/0702014
---------------------------------------------------------------
5235. HOMOGENIZATION OF PERIODIC LINEAR DEGENERATE PDES
Martin Hairer and Etienne Pardoux
It is well-known under the name of `periodic homogenization' that,
under a
centering condition of the drift, a periodic diffusion process on R^d
converges, under diffusive rescaling, to a d-dimensional Brownian
motion.
Existing proofs of this result all rely on uniform ellipticity or
hypoellipticity assumptions on the diffusion. In this paper, we
considerably
weaken these assumptions in order to allow for the diffusion
coefficient to
even vanish on an open set.
As a consequence, it is no longer the case that the effective
diffusivity
matrix is necessarily non-degenerate. It turns out that, provided
that some
very weak regularity conditions are met, the range of the effective
diffusivity
matrix can be read off the shape of the support of the invariant
measure for
the periodic diffusion. In particular, this gives some easily verifiable
conditions for the effective diffusivity matrix to be of full rank.
We also
discuss the application of our results to the homogenization of a
class of
elliptic and parabolic PDEs.
http://front.math.ucdavis.edu/math.PR/0702304
---------------------------------------------------------------
5236. A QUENCHED INVARIANCE PRINCIPLE FOR CERTAIN BALLISTIC RANDOM
WALKS IN I.I.D. ENVIRONMENTS
Noam Berger and Ofer Zeitouni
We prove that every random walk in i.i.d. environment in dimension
greater
than or equal to 4 that has an almost sure positive speed in a certain
direction, an annealed invariance principle and some mild integrability
condition for regeneration times also satisfies a quenched invariance
principle. The argument is based on intersection estimates and a
theorem of
Bolthausen and Sznitman.
http://front.math.ucdavis.edu/math.PR/0702306
---------------------------------------------------------------
5237. EXISTENCE AND SMOOTHNESS OF THE DENSITY FOR SPATIALLY
HOMOGENEOUS SPDES
David Nualart (University of Kansas) and Lluis Quer-Sardanyons
(Universitat Autonoma de Barcelona)
In this paper, we extend Walsh's stochastic integral with respect to a
Gaussian noise, white in time and with some homogeneous spatial
correlation, in
order to be able to integrate some random measure-valued processes. This
extension turns out to be equivalent to Dalang's one. Then we study
existence
and regularity of the density of the probability law for the real-
valued mild
solution to a general second order stochastic partial differential
equation
driven by such a noise. For this, we apply the techniques of the
Malliavin
calculus. Our results apply to the case of the stochastic heat
equation in any
space dimension and the stochastic wave equation in space dimension
$d=1,2,3$.
Moreover, for these particular examples, known results in the
literature have
been improved.
http://front.math.ucdavis.edu/math.PR/0702312
---------------------------------------------------------------
5238. MEASURES WITH ZEROS IN THE INVERSE OF THEIR MOMENT MATRIX
J. William Helton and Jean B. Lasserre and Mihai Putinar
We investigate and discuss when the inverse of a multivariate truncated
moment matrix of a measure $\mu$ has zeros in some prescribed
entries. We
describe precisely which pattern of these zeroes corresponds to
independence,
namely, the measure having a product structure. A more refined
finding is that
the key factor forcing a zero entry in this inverse matrix is a certain
conditional triangularity property of the orthogonal polynomials
associated
with the measure $\mu$.
http://front.math.ucdavis.edu/math.PR/0702314
---------------------------------------------------------------
5239. CONVERGENCE OF WEIGHTED POWER VARIATIONS OF FRACTIONAL BROWNIAN
MOTION
Mihai Gradinaru (IECN) and Ivan Nourdin (PMA)
The first part of the paper contains the study of the convergence for
some
weighted power variations of a fractional Brownian motion B with
Hurst index H
in (0,1). The behaviour is different when H<1/2 and powers are odd,
compared
with the case when H=1/2 or when H>1/2 and powers are even. In the
second part,
one applies the results of the first part to compute the exact rate of
convergence of some approximating schemes associated to scalar
stochastic
differential equations driven by B. The limit of the error between
the exact
solution and the considered scheme (whose size depends on the Hurst
index H) is
computed explicitly.
http://front.math.ucdavis.edu/math.PR/0702317
---------------------------------------------------------------
5240. NEIGHBOR SELECTION AND HITTING PROBABILITY IN SMALL-WORLD GRAPHS
Oskar Sandberg
Small-world graphs, which combine randomized and structured elements,
are
seen as prevalent in nature. Jon Kleinberg showed that in some graphs
of this
type it is possible to route, or navigate, between vertices in few
steps even
with very little knowledge of the graph itself.
We discuss a different criterion for graphs being navigable in
this sense,
relating the neighbor selection of a vertex with the hitting
probability of
routed walks. In several models starting from both discrete and
continuous
settings, this can be showed to lead to graphs with the desired
properties. It
also leads directly to a evolutionary model for the creation of
similar graphs
by the stepwise rewiring of the edges, and we conjecture, supported by
simulations, that these too are navigable.
http://front.math.ucdavis.edu/math.PR/0702325
---------------------------------------------------------------
5241. CONTINUITY IN LAW WITH RESPECT TO THE HURST PARAMETER OF THE
LOCAL TIME OF THE FRACTIONAL BROWNIAN MOTION
Maria Jolis and No\`elia Viles
We give a result of stability in law of the local time of the fractional
Brownian motion with respect to small perturbations of the Hurst
parameter.
Concretely, we prove that the law (in the space of continuous
functions) of the
local time of the fractional Brownian motion with Hurst parameter $H$
converges
weakly to that of the local time of $B^{H_0}$, when $H$ tends to $H_0$.
http://front.math.ucdavis.edu/math.PR/0702330
---------------------------------------------------------------
5242. TIGHTNESS CONDITIONS FOR POLYMER MEASURES
Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
We give sufficient conditions for tightness in the space C([0,1]) for
sequences of probability measures which enjoy a suitable decoupling
between
zero level set and excursions. Applications of our results are given
in the
context of (homogeneous, periodic and disordered) random walk models for
polymers and interfaces.
http://front.math.ucdavis.edu/math.PR/0702331
---------------------------------------------------------------
5243. A NOTE ON EQUILIBRIUM GLAUBER AND KAWASAKI DYNAMICS FOR FERMION
POINT PROCESSES
E. Lytvynov and N. Ohlerich
We construct two types of equilibrium dynamics of infinite particle
systems
in a locally compact Polish space $X$, for which certain fermion point
processes are invariant. The Glauber dynamics is a birth-and-death
process in
$X$, while in the case of the Kawasaki dynamics interacting particles
randomly
hop over $X$. We establish conditions on generators of both dynamics
under
which corresponding conservative Markov processes exist.
http://front.math.ucdavis.edu/math.PR/0702338
---------------------------------------------------------------
5244. MULTIPLICATIVE FREE CONVOLUTION AND INFORMATION-PLUS-NOISE TYPE
MATRICES
{\O}yvind Ryan and M\'erouane Debbah
Free probability and random matrix theory has shown to be a fruitful
combination in many fields of research, such as digital
communications, nuclear
physics and mathematical finance. The link between free probability and
eigenvalue distributions of random matrices will be strengthened
further in
this paper. It will be shown how the concept of multiplicative free
convolution
can be used to express known results for eigenvalue distributions of
a type of
random matrices called Information-Plus-Noise matrices. The result is
proved in
a free probability framework, and some new results, useful for
problems related
to free probability, are presented in this context. The connection
between free
probability and estimators for covariance matrices is also made
through the
notion of free deconvolution.
http://front.math.ucdavis.edu/math.PR/0702342
---------------------------------------------------------------
5245. LAW OF LARGE NUMBERS AND CENTRAL LIMIT THEOREM UNDER NONLINEAR
EXPECTATIONS
Shige Peng
The law of large numbers (LLN) and central limit theorem (CLT) are
long and
widely been known as two fundamental results in probability theory.
Recently problems of model uncertainties in statistics, measures
of risk and
superhedging in finance motivated us to introduce, in [4] and [5]
(see also
[2], [3] and references herein), a new notion of sublinear
expectation, called
\textquotedblleft% $G$-expectation\textquotedblright, and the related
\textquotedblleft$G$-normal distribution\textquotedblright from which
we were
able to define G-Brownian motion as well as the corresponding stochastic
calculus. The notion of G-normal distribution plays the same
important rule in
the theory of sublinear expectation as that of normal distribution in
the
classic probability theory. It is then natural and interesting to ask
if we
have the corresponding LLN and CLT under a sublinear expectation and, in
particular, if the corresponding limit distribution of the CLT is a G-
normal
distribution. This paper gives an affirmative answer. The proof of
our CLT is
short since we borrow a deep interior estimate of fully nonlinear PDE
in [6]
which extended a profound result of [1] (see also [7]) to parabolic
PDEs. The
assumptions of our LLN and CLT can be still improved. But the discovered
phenomenon plays the same important rule in the theory of nonlinear
expectation
as that of the classical LLN and CLT in classic probability theory.
http://front.math.ucdavis.edu/math.PR/0702358
---------------------------------------------------------------
5246. SMOOTHNESS OF DENSITY FOR SOLUTIONS TO STOCHASTIC DIFFERENTIAL
EQUATIONS WITH JUMPS
T.R.Cass
We consider a solution to a generic stochastic differential equation
with
jumps and show that for each time the marginal law of the solution
has an
infinitely differentiable density with respect to Lebesgue measure
under a
uniform version of Hoermanders conditions. Our results are proved
subject to
some restrictions on the rate of growth of the jump measure near zero
and are
accomplished using developments of traditional arguments in Malliavin
calculus.
A key ingredient in our proof is a generalisation of Norris
semimartingale
inequality to discontinuous semimartingales. Unlike previous work,
our results
extend beyond the case finite activity jump processes.
http://front.math.ucdavis.edu/math.PR/0702364
---------------------------------------------------------------
5247. A POPULATION MODEL FOR $\LAMBDA$-COALESCENTS WITH NEUTRAL
MUTATIONS
Andreas Nordvall Lager{\aa}s
Bertoin and Le Gall (2003) introduced a certain probability measure
valued
Markov process that describes the evolution of a population, such
that a sample
from this population would exhibit a genealogy given by the so-called
$\Lambda$-coalescent, or coalescent with multiple collisions, introduced
independently by Pitman (1999) and Sagitov (1999). We show how this
process can
be extended to the case where lineages can experience mutations.
Regenerative
compositions enter naturally into this model, which is somewhat
surprising,
considering a negative result by M{\"o}hle (2007).
http://front.math.ucdavis.edu/math.PR/0702367
---------------------------------------------------------------
5248. INTEGRAL EQUATIONS IN THE THEORY OF LEVY PROCESSES
Lev Sakhnovich
In this article we consider the Levy processes and the corresponding
semigroup. We represent the generator of this semigroup in a
convolution form.
Using the obtained convolution form and the theory of integral
equations we
investigate the properties of a wide class of Levy processes (potential,
quasi-potential, the probability of the Levy process remaining within
the given
domain, long time behavior, stable processes). We analyze in detail a
number of
concrete examples of the Levy processes (stable processes, the
variance damped
Levy processes, the variance gamma processes, the normal Gaussian
process, the
Meixner process, the compound Poisson process).
http://front.math.ucdavis.edu/math.PR/0702378
---------------------------------------------------------------
5249. ON THE CIRCULAR LAW
F. G\"otze and A. Tikhomirov
We consider the joint distribution of real and imaginary parts of
eigenvalues
of random matrices with independent real entries with mean zero and unit
variance. We prove the convergence of this distribution to the uniform
distribution on the unit disc without assumptions on the existence of
a density
for the distribution of entries. We assume however that the entries have
sub-Gaussian tails or are sparsely non-zero.
http://front.math.ucdavis.edu/math.PR/0702386
---------------------------------------------------------------
5250. STATIONARY FLOWS AND UNIQUENESS OF INVARIANT MEASURES
Francois Baccelli and Takis Konstantopoulos
In this short paper, we consider a quadruple $(\Omega, \AA, \theta,
\mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $
\theta$ is
a measurable bijection from $\Omega$ into itself that preserves the
measure
$\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained
by taking
cycles (excursions) of iterates of $\theta$ from $B$. We then derive
a relation
for $\mu_B$ that involves the forward and backward hitting times of $B
$ by the
trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$.
Although classical in appearance, its use in obtaining uniqueness of
invariant
measures of various stochastic models seems to be new. We apply the
concept to
countable Markov chains and Harris processes.
http://front.math.ucdavis.edu/math.PR/0702391
---------------------------------------------------------------
5251. MAJORITY BOOTSTRAP PERCOLATION ON THE HYPERCUBE
J\'ozsef Balogh and B\'ela Bollob\'as and Robert Morris
In majority bootstrap percolation on a graph G, an infection spreads
according to the following deterministic rule: if at least half of the
neighbours of a vertex v are already infected, then v is also
infected, and
infected vertices remain infected forever. Percolation occurs if
eventually
every vertex is infected.
The elements of the set of initially infected vertices, A \subset V
(G), are
normally chosen independently at random, each with probability p,
say. This
process has been extensively studied on the sequence of torus graphs
[n]^d, for
n = 1,2,..., where d = d(n) is either fixed or a very slowly growing
function
of n. For example, Cerf and Manzo showed that the critical
probability is o(1)
if d(n) < log*(n), i.e., if p = p(n) is bounded away from zero then the
probability of percolation on [n]^d tends to one as n goes to infinity.
In this paper we study the case when the growth of d to infinity
is not
excessively slow; in particular, we show that the critical
probability is 1/2 +
o(1) if d > (loglog(n))^2 logloglog(n), and give much stronger bounds
in the
case that G is the hypercube, [2]^d.
http://front.math.ucdavis.edu/math.CO/0702373
---------------------------------------------------------------
5252. BERGMAN KERNELS AND WEIGHTED EQUILIBRIUM MEASURES OF C^N
Robert Berman
We obtain various convergence results for the Bergman kernel of the
Hilbert
space of all polynomials in \C^{n} of total degree at most k,
equipped with a
weighted norm. The weight function is assumed to be a smooth function
in \C^{n}
which grows faster than the logarithm of the squared distance
function. The
convergence is studied in the large k limit and is expressed in terms
of the
global equilibrium potential associated to the weight function, as
well as in
terms of the Monge-Ampere measure of the weight function itself on a
certain
bounded support set S. These results apply directly to the study of the
distribution of zeroes of random polynomials and of the eigenvalues
of random
normal matrices.
http://front.math.ucdavis.edu/math.CV/0702357
---------------------------------------------------------------
5253. ON THE NUMBER OF MINIMA OF A RANDOM POLYNOMIAL
Jean-Pierre Dedieu and Gregorio Malajovich
We give an upper bound in O(d ^((n+1)/2)) for the number of critical
points
of a normal random polynomial. The number of minima (resp. maxima) is in
O(d^((n+1)/2)) P_n, where P_n is the (unknown) measure of the set of
symmetric
positive matrices in the Gaussian Orthogonal Ensemble GOE(n).
Finally, we give
a closed form expression for the number of maxima (resp. minima) of a
random
univariate polynomial, in terms of hypergeometric functions.
http://front.math.ucdavis.edu/math.NA/0702360
---------------------------------------------------------------
5254. DIFFUSION APPROXIMATIONS FOR CONTROLLED STOCHASTIC NETWORKS:
AN ASYMPTOTIC BOUND FOR THE VALUE FUNCTION
Amarjit Budhiraja and Arka Prasanna Ghosh
We consider the scheduling control problem for a family of unitary
networks
under heavy traffic, with general interarrival and service times,
probabilistic
routing and infinite horizon discounted linear holding cost. A natural
nonanticipativity condition for admissibility of control policies is
introduced. The condition is seen to hold for a broad class of
problems. Using
this formulation of admissible controls and a time-transformation
technique, we
establish that the infimum of the cost for the network control
problem over all
admissible sequencing control policies is asymptotically bounded
below by the
value function of an associated diffusion control problem (the
Brownian control
problem). This result provides a useful bound on the best achievable
performance for any admissible control policy for a wide class of
networks.
http://front.math.ucdavis.edu/math.PR/0702402
---------------------------------------------------------------
5255. FUNCTIONAL INEQUALITIES AND UNIQUENESS OF THE GIBBS MEASURE --
FROM LOG-SOBOLEV TO POINCAR\'{E}
Pierre-Andr\'{e} Zitt (MODAL'X)
In a statistical mechanics model with unbounded spins, we prove
uniqueness of
the Gibbs measure under various assumptions on finite volume functional
inequalities. We follow the approach of G. Royer (1999) and obtain
uniqueness
by showing convergence properties of a Glauber-Langevin dynamics. The
result
was known when the measures on the box $[-n,n]^d$ (with free boundary
conditions) satisfied the same logarithmic Sobolev inequality. We
generalize
this in two directions: either the constants may be allowed to grow
sub-linearly in the diameter, or we may suppose a weaker inequality than
log-Sobolev, but stronger than Poincar\'{e}. We conclude by giving a
heuristic
argument showing that this could be the right inequalities to look at.
http://front.math.ucdavis.edu/math.PR/0702403
---------------------------------------------------------------
5256. BOUNDED SOLUTIONS TO BACKWARD SDE'S WITH JUMPS FOR UTILITY
OPTIMIZATION AND INDIFFERENCE HEDGING
Dirk Becherer
We prove results on bounded solutions to backward stochastic
equations driven
by random measures. Those bounded BSDE solutions are then applied to
solve
different stochastic optimization problems with exponential utility
in models
where the underlying filtration is noncontinuous. This includes
results on
portfolio optimization under an additional liability and on dynamic
utility
indifference valuation and partial hedging in incomplete financial
markets
which are exposed to risk from unpredictable events. In particular, we
characterize the limiting behavior of the utility indifference
hedging strategy
and of the indifference value process for vanishing risk aversion.
http://front.math.ucdavis.edu/math.PR/0702405
---------------------------------------------------------------
5257. THE CHOQUET-DENY THEOREM AND DISTAL PROPERTIES OF TOTALLY
DISCONNECTED LOCALLY COMPACT GROUPS OF POLYNOMIAL GROWTH
W. Jaworski and C. R. E. Raja
We obtain sufficient and necessary conditions for the Choquet-Deny
theorem to
hold in the class of compactly generated totally disconnected locally
compact
groups of polynomial growth, and in a larger class of totally
disconnected
generalized $\ov{FC}$-groups. The following conditions turn out to be
equivalent when $G$ is a metrizable compactly generated totally
disconnected
locally compact group of polynomial growth: (i) the Choquet-Deny
theorem holds
for $G$; (ii) the group of inner automorphisms of $G$ acts distally
on $G$;
(iii) every inner automorphism of $G$ is distal; (iv) the contraction
subgroup
of every inner automorphism of $G$ is trivial; (v) $G$ is a SIN
group. We also
show that for every probability measure $\mu$ on a totally disconnected
compactly generated locally compact second countable group of polynomial
growth, the Poisson boundary is a homogeneous space of $G$, and that
it is a
compact homogeneous space when the support of $\mu$ generates $G$.
http://front.math.ucdavis.edu/math.PR/0702407
---------------------------------------------------------------
5258. MARKET FREE LUNCH AND LARGE FINANCIAL MARKETS
Irene Klein
The main result of the paper is a version of the fundamental theorem
of asset
pricing (FTAP) for large financial markets based on an asymptotic
concept of no
market free lunch for monotone concave preferences. The proof uses
methods from
the theory of Orlicz spaces. Moreover, various notions of no asymptotic
arbitrage are characterized in terms of no asymptotic market free
lunch; the
difference lies in the set of utilities. In particular, it is shown
directly
that no asymptotic market free lunch with respect to monotone concave
utilities
is equivalent to no asymptotic free lunch. In principle, the paper
can be seen
as the large financial market analogue of [Math. Finance 14 (2004)
351--357]
and [Math. Finance 16 (2006) 583--588].
http://front.math.ucdavis.edu/math.PR/0702409
---------------------------------------------------------------
5259. SEPARATION CUT-OFFS FOR BIRTH AND DEATH CHAINS
Persi Diaconis and Laurent Saloff-Coste
This paper gives a necessary and sufficient condition for a sequence
of birth
and death chains to converge abruptly to stationarity, that is, to
present a
cut-off. The condition involves the notions of spectral gap and
mixing time. Y.
Peres has observed that for many families of Markov chains, there is
a cut-off
if and only if the product of spectral gap and mixing time tends to
infinity.
We establish this for arbitrary birth and death chains in continuous
time when
the convergence is measured in separation and the chains all start at 0.
http://front.math.ucdavis.edu/math.PR/0702411
---------------------------------------------------------------
5260. HARRIS RECURRENCE OF METROPOLIS-WITHIN-GIBBS AND TRANS-
DIMENSIONAL MARKOV CHAINS
Gareth O. Roberts and Jeffrey S. Rosenthal
A $\phi$-irreducible and aperiodic Markov chain with stationary
probability
distribution will converge to its stationary distribution from almost
all
starting points. The property of Harris recurrence allows us to replace
``almost all'' by ``all,'' which is potentially important when
running Markov
chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings
algorithms
are known to be Harris recurrent. In this paper, we consider
conditions under
which Metropolis-within-Gibbs and trans-dimensional Markov chains are
or are
not Harris recurrent. We present a simple but natural two-dimensional
counter-example showing how Harris recurrence can fail, and also a
variety of
positive results which guarantee Harris recurrence. We also present
some open
problems. We close with a discussion of the practical implications
for MCMC
algorithms.
http://front.math.ucdavis.edu/math.PR/0702412
---------------------------------------------------------------
5261. SENSITIVITY ANALYSIS OF UTILITY-BASED PRICES AND RISK-TOLERANCE
WEALTH PROCESSES
Dmitry Kramkov and Mihai S\^{{\i}}rbu
In the general framework of a semimartingale financial model and a
utility
function $U$ defined on the positive real line, we compute the first-
order
expansion of marginal utility-based prices with respect to a
``small'' number
of random endowments. We show that this linear approximation has some
important
qualitative properties if and only if there is a risk-tolerance
wealth process.
In particular, they hold true in the following polar cases:
\begin{tabular}@p97mm@ for any utility function $U$, if and only if
the set of
state price densities has a greatest element from the point of view of
second-order stochastic dominance;for any financial model, if and
only if $U$
is a power utility function ($U$ is an exponential utility function
if it is
defined on the whole real line). \end{tabular}
http://front.math.ucdavis.edu/math.PR/0702413
---------------------------------------------------------------
5262. CENTRAL LIMIT THEOREM FOR THE ON-LINE NEAREST-NEIGHBOUR GRAPH
Andrew R. Wade
The on-line nearest-neighbour graph on a sequence of uniform random
points in
$(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest
neighbour amongst its predecessors. For the total power-weighted edge
length of
this graph, with weight exponent $\alpha \in (0,d/2)$, we prove a
central limit
theorem (in the large-sample limit), including an expression for the
limiting
variance. In contrast, we give a convergence result (with no scaling)
for
$\alpha > d/2$. Both these results extend previous work. We also make
some
progress in the critical case $\alpha=d/2$.
http://front.math.ucdavis.edu/math.PR/0702414
---------------------------------------------------------------
5263. THE MEAN, VARIANCE AND LIMITING DISTRIBUTION OF TWO STATISTICS
SENSITIVE TO PHYLOGENETIC TREE BALANCE
Michael G. B. Blum and Olivier Fran\c{c}ois and Svante Janson
For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this
article,
this statistic is studied under the Yule and uniform model of
phylogenetic
trees. The main tool of analysis is a coupling argument with another
well-known
index called the Sackin statistic. Asymptotics for the mean, variance
and
covariance of these two statistics are obtained, as well as their
limiting
joint distribution for large phylogenies. Under the Yule model, the
limiting
distribution arises as a solution of a functional fixed point
equation. Under
the uniform model, the limiting distribution is the Airy
distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and
the
Catalan models for binary search trees.
http://front.math.ucdavis.edu/math.PR/0702415
---------------------------------------------------------------
5264. EXISTENCE OF OPTIMAL CONTROLS FOR SINGULAR CONTROL PROBLEMS
WITH STATE CONSTRAINTS
Amarjit Budhiraja and Kevin Ross
We establish the existence of an optimal control for a general class of
singular control problems with state constraints. The proof uses weak
convergence arguments and a time rescaling technique. The existence
of optimal
controls for Brownian control problems \citehar, associated with a
broad family
of stochastic networks, follows as a consequence.
http://front.math.ucdavis.edu/math.PR/0702418
---------------------------------------------------------------
5265. STATIONARITY AND GEOMETRIC ERGODICITY OF A CLASS OF NONLINEAR
ARCH MODELS
Youssef Sa\"{{\i}}di and Jean-Michel Zako\"{{\i}}an
A class of nonlinear ARCH processes is introduced and studied. The
existence
of a strictly stationary and $\beta$-mixing solution is established
under a
mild assumption on the density of the underlying independent process.
We give
sufficient conditions for the existence of moments. The analysis
relies on
Markov chain theory. The model generalizes some important features of
standard
ARCH models and is amenable to further analysis.
http://front.math.ucdavis.edu/math.PR/0702419
---------------------------------------------------------------
5266. CORRECTIONS AND ACKNOWLEDGMENT FOR ``LOCAL LIMIT THEORY AND
LARGE DEVIATIONS FOR SUPERCRITICAL BRANCHING PROCESSES''
P. E. Ney and Anand N. Vidyashankar
Corrections and acknowledgment for ``Local limit theory and large
deviations
for supercritical branching processes'' [math.PR/0407059]
http://front.math.ucdavis.edu/math.PR/0702421
---------------------------------------------------------------
5267. CORRECTION. ERROR ESTIMATES FOR BINOMIAL APPROXIMATIONS OF GAME
OPTIONS
Yuri Kifer
Correction for Error estimates for binomial approximations of game
options
[math.PR/0607123]
http://front.math.ucdavis.edu/math.PR/0702423
---------------------------------------------------------------
5268. NETWORK-BASED ANALYSIS OF STOCHASTIC SIR EPIDEMIC MODELS WITH
RANDOM AND PROPORTIONATE MIXING
Eben Kenah and James Robins
In this paper, we outline the theory of percolation networks and
their use in
the analysis of stochastic epidemic models on undirected contact
networks. We
then show how the same theory can be used to analyze epidemic models
with
random mixing. In the percolation network for a random-mixing model,
undirected
edges disappear in the limit of a large population, so the
percolation network
is purely directed. In a series of simulations, we show that percolation
networks accurately predict the mean outbreak size and probability
and final
size of an epidemic for a variety of epidemic models in homogeneous and
heterogeneous populations. Finally, we show conditions under which
percolation
network models are equivalent to branching processes and use percolation
networks to re-derive several classical results from different areas of
infectious disease epidemiology. In an appendix, we show how percolation
networks can be defined for any time-homogeneous stochastic epidemic
model. We
conclude that the theory of percolation on semi-directed networks
provides a
very general framework for the analysis of stochastic SIR epidemic
models in
closed populations, which are an important part of theoretical
infectious
disease epidemiology.
http://front.math.ucdavis.edu/q-bio.QM/0702027
---------------------------------------------------------------
5269. LIMITING SHAPES FOR DETERMINISTIC INTERNAL GROWTH MODELS
Anne Fey and Frank Redig
We study the rotor router model and two deterministic sandpile
models. For
the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved
that the
limiting shape of the growth cluster is a sphere. For the other two
models,
only bounds in dimension 2 are known. A unified approach for these
models with
a new parameter $h$ (the initial number of particles at each site),
allows to
prove a number of new limiting shape results in any dimension $d \geq
1$.
For the rotor router model, the limiting shape is a sphere for all
values of
$h$. For one of the sandpile models, and $h=2d-2$ (the maximal
value), the
limiting shape is a cube. For both sandpile models, the limiting
shape is a
sphere in the limit $h \to -\infty$. Finally, we prove that the rotor
router
shape contains a diamond, which is a new result even in the case
studied by
Levine and Peres.
http://front.math.ucdavis.edu/math.PR/0702450
---------------------------------------------------------------
5270. CONVEXITY THEORY FOR THE TERM STRUCTURE EQUATION
Erik Ekstrom and Johan Tysk
We study convexity and monotonicity properties for prices of bonds
and bond
options when the short rate is modeled by a diffusion process. We
provide
conditions under which convexity of the price in the short rate is
guaranteed.
Under these conditions the price is decreasing in the drift and
increasing in
the volatility of the short rate. We also study convexity properties
of the
logarithm of the price.
http://front.math.ucdavis.edu/math.AP/0702435
---------------------------------------------------------------
5271. SOME APPLICATIONS AND METHODS OF LARGE DEVIATIONS IN FINANCE
AND INSURANCE
Huyen Pham (PMA)
In these notes, we present some methods and applications of large
deviations
to finance and insurance. We begin with the classical ruin problem
related to
the Cramer's theorem and give en extension to an insurance model with
investment in stock market. We then describe how large deviation
approximation
and importance sampling are used in rare event simulation for option
pricing.
We finally focus on large deviations methods in risk management for the
estimation of large portfolio losses in credit risk and portfolio
performance
in market investment.
http://front.math.ucdavis.edu/math.PR/0702473
---------------------------------------------------------------
5272. A NOTE ON PERCOLATION ON \Z^D: ISOPERIMETRIC PROFILE VIA
EXPONENTIAL CLUSTER REPULSION
Gabor Pete
We show that for all p>p_c(\Z^d) percolation parameters, the
probability that
the cluster of the origin is finite but has at least t vertices at
distance one
from the infinite cluster is exponentially small in t. Then we use
this to give
a very short proof of the important fact that the isoperimetric
profile of the
infinite cluster basically coincides with the profile of the original
lattice.
This implies for instance that simple random walk on the largest
cluster of a
finite box [-n,n]^d with high probability has L^\infty-mixing time
\Theta(n^2),
and that the heat kernel (return probability) on the infinite cluster
a.s.
decays like p_n(o,o)=O(n^{-d/2}). Versions of these results have been
proven by
Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004)
and Rau
(2006). We also give a short proof of a theorem of Angel, Benjamini,
Berger and
Peres (2006): the infinite percolation cluster of a wedge in \Z^3 is
a.s.
transient whenever the wedge itself is transient.
http://front.math.ucdavis.edu/math.PR/0702474
---------------------------------------------------------------
5273. CENTRAL LIMIT THEOREM FOR A CLASS OF RELATIVISTIC DIFFUSIONS
J\"{u}rgen Angst (IRMA) and Jacques Franchi (IRMA)
Two similar Minkowskian diffusions have been considered, on one hand by
Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR],
[DR]),
and on the other hand by Dunkel and H\"{a}nggi ([DH1], [DH2]). We
address here
the question, asked in ([DH1], [DH2]), of the asymptotic behaviour of
the
variance of such diffusions. More generally, we establish a central
limit
theorem for a class of Minkowskian diffusions, to which the two above
ones
belong. As a consequence, we correct a partially wrong guess in [DH1].
http://front.math.ucdavis.edu/math.PR/0702481
---------------------------------------------------------------
5274. SCALING LIMITS FOR GRADIENT SYSTEMS IN RANDOM ENVIRONMENT
P. Goncalves and M.D. Jara
For interacting particle systems that satisfies the gradient
condition, the
hydrodynamic limit and the equilibrium fluctuations are well known.
We prove
that under the presence of a symmetric random environment, these
scaling limits
also hold for almost every choice of the environment, with homogenized
coefficients that does not depend on the particular realization of
the random
environment.
http://front.math.ucdavis.edu/math.PR/0702513
---------------------------------------------------------------
5275. FIRST HITTING TIME AND PLACE, MONOPOLES AND MULTIPOLES FOR
PSEUDO-PROCESSES DRIVEN BY THE EQUATION $\PARTIAL/\PARTIAL T =
\PM\PARTIAL^N/\PARTIAL X^N$
Aim\'e Lachal
Consider the high-order heat-type equation $\partial u/\partial
t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce
the related
Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several
functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and
minimum $m(t)$
up to time $t$; the hitting times $\tau_a^+$ and $\tau_a^-$ of the
half lines
$(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit
expressions
for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$,
as well
as those of the vectors $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X
(\tau_a^-))$.
http://front.math.ucdavis.edu/math.PR/0702541
---------------------------------------------------------------
5276. DYNAMICS FOR THE BROWNIAN WEB AND THE EROSION FLOW
Chris Howitt and Jon Warren
The Brownian web is a random object that occurs as the scaling limit
of an
infinite system of coalescing random walks. Perturbing this system of
random
walks by, independently at each point in space-time, resampling the
random walk
increments, leads to some natural dynamics. In this paper we consider
the
corresponding dynamics for the Brownian web. In particular, pairs of
coupled
Brownian webs are studied, where the second web is obtained from the
first by
perturbing according to these dynamics. A stochastic flow of kernels,
which we
call the erosion flow, is obtained via a filtering construction from
such
coupled Brownian webs, and the N-point motions of this flow of
kernels are
identified.
http://front.math.ucdavis.edu/math.PR/0702542
---------------------------------------------------------------
5277. EXTREME POINTS OF THE CONVEX SET OF JOINT PROBABILITY
DISTRIBUTIONS WITH FIXED MARGINALS
K. R. Parthasarathy
By using a quantum probabilistic approach we obtain a description of the
extreme points of the convex set of all joint probability
distributions on the
product of two standard Borel spaces with fixed marginal distributions.
http://front.math.ucdavis.edu/math.PR/0702544
---------------------------------------------------------------
5278. QQ PLOTS, RANDOM SETS AND DATA FROM A HEAVY TAILED DISTRIBUTION
Bikramjit Das and Sidney I. Resnick
The QQ plot is a commonly used technique for informally deciding
whether a
univariate random sample of size n comes from a specified
distribution F. The
QQ plot graphs the sample quantiles against the theoretical quantiles
of F and
then a visual check is made to see whether or not the points are
close to a
straight line. For a location and scale family of distributions, the
intercept
and slope of the straight line provide estimates for the shift and scale
parameters of the distribution respectively. Here we consider the set
S_n of
points forming the QQ plot as a random closed set in R^2. We show
that under
certain regularity conditions on the distribution F, S_n converges in
probability to a closed, non-random set. In the heavy tailed case
where 1-F is
a regularly varying function, a similar result can be shown but a
modification
is necessary to provide a statistically sensible result since
typically F is
not completely known.
http://front.math.ucdavis.edu/math.PR/0702551
---------------------------------------------------------------
5279. VARIANCE ASYMPTOTICS AND CENTRAL LIMIT THEOREMS FOR GENERALIZED
GROWTH PROCESSES WITH APPLICATIONS TO CONVEX HULLS AND MAXIMAL POINTS
Tomasz Schreiber and Joseph E. Yukich
We show that the random point measures induced by vertices in the
convex hull
of a Poisson sample on the unit ball, when properly scaled and centered,
converge to those of a mean zero Gaussian field. We establish
limiting variance
and covariance asymptotics in terms of the density of the Poisson
sample.
Similar results hold for the point measures induced by the maximal
points in a
Poisson sample. The approach involves introducing a generalized
spatial birth
growth process allowing for cell overlap.
http://front.math.ucdavis.edu/math.PR/0702553
---------------------------------------------------------------
5280. AN IMPROVED METHOD FOR MODEL SELECTION BASED ON INFORMATION
CRITERIA
Guilhem Coq (1) and Olivier Alata (2) and Marc Arnaudon (1) and
Christian Olivier (2) ((1) Laboratoire de Math\'ematiques et
Applications Poitiers
France, (2) Laboratoire Signal Image et Communications Poitiers
France)
Information criteria are an appropriate and widely used tool for solving
model selection problems. However, different ways to use them exist,
each
leading to a more or less precise approximation of the sought model.
In this
paper, we mainly present two methods of utilisation of information
criteria :
the classical one which is generally used and an alternative one,
more precise
but requiring a little more calculations. Those methods are compared
on 1-D and
2-D autoregressive models ; we use a synthetized process for the 1-D
case and
texture images for the 2-D case. We also work with the original phi_beta
criterion which includes all others usual criteria such as AIC, BIC,
and phi.
http://front.math.ucdavis.edu/math.ST/0702540
---------------------------------------------------------------
5281. A STOCHASTIC LAGRANGIAN PROOF OF GLOBAL EXISTENCE OF THE NAVIER-
STOKES EQUATIONS FOR FLOWS WITH SMALL REYNOLDS NUMBER
Gautam Iyer
We consider the incompressible Navier-Stokes equations with spatially
periodic boundary conditions. If the Reynolds number is small enough
we provide
an elementary short proof of the existence of global in time H\"older
continuous solutions. Our proof is based on the stochastic Lagrangian
formulation of the Navier-Stokes equations, and works in both the two
and three
dimensional situation.
http://front.math.ucdavis.edu/math.AP/0702506
---------------------------------------------------------------
5282. LARGE DEVIATION ESTIMATES OF THE CROSSING PROBABILITY FOR
PINNED GAUSSIAN PROCESSES
L. Caramellino and B. Pacchiarotti
The paper deals with the asymptotic behavior of the bridge of a Gaussian
process conditioned to stay in $n$ fixed points at $n$ fixed past
instants. In
particular, functional large deviation results are stated for small
time.
Several examples are considered: integrated or not fractional
Brownian motion,
$m$-fold integrated Brownian motion. As an application, the
asymptotic behavior
of the exit probability is studied and used for the practical purpose
of the
numerical computation, via Monte Carlo methods, of the hitting
probability up
to a given time.
http://front.math.ucdavis.edu/math.PR/0702573
---------------------------------------------------------------
5283. WELL-POSEDNESS AND INVARIANT MEASURES FOR HJM MODELS WITH
DETERMINISTIC VOLATILITY AND LEVY NOISE
Carlo Marinelli
We give sufficient conditions for existence, uniqueness and
ergodicity of
invariant measures for Musiela's stochastic partial differential
equation with
deterministic volatility and a Hilbert space valued driving Levy noise.
Conditions for the absence of arbitrage and for the existence of mild
solutions
are also discussed.
http://front.math.ucdavis.edu/math.PR/0702622
---------------------------------------------------------------
5284. THE RIFF-SHUFFLE DISTRIBUTION IS UNIMODAL
S. Gerhold
We show that the probability mass function of the riff-shuffle
distribution,
also known as the minimum negative binomial distribution, is
unimodal, but in
general not log-concave.
http://front.math.ucdavis.edu/math.PR/0702639
---------------------------------------------------------------
5285. BILATERAL CANONICAL CASCADES: MULTIPLICATIVE REFINEMENT PATHS
TO WIENER'S AND VARIANT FRACTIONAL BROWNIAN LIMITS
Julien Barral and Benoit Mandelbrot
The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b
\geq
2$%), and $p\in (0,1)$ is a parameter. The first construction stage
divides the
unit interval into $b$ subintervals and multiplies the density in each
subinterval by either 1 or -1 with the respective frequencies of $
\frac{1%
}{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the
resulting
density can be renormalized so that, as $n\to \infty $ ($n$ being the
number of
iterations) the signed measure converges in some sense to a non-
degenerate
limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$,
renormalization
creates a martingale, the convergence is strong, and the limit shares
the
H\"{o}lder and Hausdorff properties of the fractional Brownian motion of
exponent $H$. If $H\leq {1}/{2}$, hence $p\leq b^{{-1}/{2}%}$, this
martingale
does not converge. However, a different normalization can be applied,
for
$H\leq {1/2}$ to the martingale itself and for $H>% {1/2}$ to the
discrepancy
between the limit and a finite approximation. In all cases the resulting
process is found to converge weakly to the Wiener Brownian motion,
independently of $H$ and of $b$. Thus, to the usual additive paths
toward
Wiener measure, this procedure adds an infinity of multiplicative paths.
http://front.math.ucdavis.edu/math.PR/0702644
---------------------------------------------------------------
5286. A SURVEY OF CONFORMALLY INVARIANT MEASURES ON H^M(\DELTA)
Doug Pickrell
The universal covering of the group PSU(1,1) acts naturally on H^m
(\delta),
the space of holomorphic differentials of order m on the Poincare
disk. The
purpose of this paper is to survey, as broadly as I am able, the
basic sources
and examples of invariant measures for this action.
http://front.math.ucdavis.edu/math.PR/0702672
---------------------------------------------------------------
5287. AR AND MA REPRESENTATION OF PARTIAL AUTOCORRELATION FUNCTIONS,
WITH APPLICATIONS
Akihiko Inoue
We prove a representation of the partial autocorrelation function
(PACF), or
the Verblunsky coefficients, of a stationary process in terms of the
AR and MA
coefficients. We apply it to show the asymptotic behaviour of the
PACF. We also
propose a new definition of short and long memory in terms of the PACF.
http://front.math.ucdavis.edu/math.SP/0702648
---------------------------------------------------------------
5288. CLASSICAL DILATIONS \`A LA QUANTUM PROBABILITY OF MARKOV
EVOLUTIONS IN DISCRETE TIME
M. Gregoratti
We study the Classical Probability analogue of the dilations of a
quantum
dynamical semigroup in Quantum Probability. Given a (not necessarily
homogeneous) Markov chain in discrete time in a finite state space E, we
introduce a second system, an environment, and a deterministic
invertible
time-homogeneous global evolution of the system E with this
environment such
that the original Markov evolution of E can be realized by a proper
choice of
the initial random state of the environment. We also compare this
dilations
with the dilations of a quantum dynamical semigroup in Quantum
Probability:
given a classical Markov semigroup, we show that it can be extended to a
quantum dynamical semigroup for which we can find a quantum dilation
to a group
of *-automorphisms admitting an invariant abelian subalgebra where
this quantum
dilation gives just our classical dilation.
http://front.math.ucdavis.edu/math.PR/0702690
---------------------------------------------------------------
5289. SOME EXTENSIONS OF FRACTIONAL BROWNIAN MOTION AND SUB-
FRACTIONAL BROWNIAN MOTION RELATED TO PARTICLE SYSTEMS
Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
In this paper we study three self-similar, long-range dependence,
Gaussian
processes. The first one, with covariance
\int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1<b
\leq 1,
|b|\leq 1+a, corresponds to fractional Brownian motion for a=0,
-1<b<1. The
second one, with covariance (2-h)(s^h+t^h-[(s+t)^h +|s-t|^h]/2),
parameter
0<h\leq 4, corresponds to sub-fractional Brownian motion for 0<h<2.
The third
one, with covariance -(s^2\log s + t^2\log t -[(s+t)^2 \log (s+t) +(s-
t)^2 \log
|s-t|]/2), is related to the second one. These processes come from
occupation
time fluctuations of certain particle systems for some values of the
parameters.
http://front.math.ucdavis.edu/math.PR/0702708
---------------------------------------------------------------
5290. HITTING PROBABILITIES FOR SYSTEMS OF NON-LINEAR STOCHASTIC
HEAT EQUATIONS WITH ADDITIVE NOISE
Robert C. Dalang and Davar Khoshnevisan and and Eulalia Nualart
We consider a system of $d$ coupled non-linear stochastic heat
equations in
spatial dimension 1 driven by $d$-dimensional additive space-time
white noise.
We establish upper and lower bounds on hitting probabilities of the
solution
$\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of
respectively
Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff
dimensions of level sets and their projections. A result of independent
interest is an anisotropic form of the Kolmogorov continuity theorem.
http://front.math.ucdavis.edu/math.PR/0702710
---------------------------------------------------------------
5291. UNIFORM IN BANDWIDTH CONSISTENCY OF CONDITIONAL U-STATISTICS
J. Dony and D. M. Mason
In 1991 Stute introduced a class of estimators called conditional
U-statistics. They can be seen as a generalization of the Nadaraya-
Watson
estimator, and their strong pointwise consistency to the general
regression
function has been obtained in the same paper by Stute. Very recently,
Gine and
Mason introduced the notion of a local U-process, which generalizes
that of a
local empirical process, and obtained central limit theorems and laws
of the
iterated logarithm for this class. We apply the methods developed by
Einmahl
and Mason (2005) and Gine and Mason (2007a,b) to establish uniform in
bandwidth
consistency to the general regression function of the estimator
proposed by
Stute.
http://front.math.ucdavis.edu/math.ST/0702696
---------------------------------------------------------------
5292. ON THE VOLUME OF NODAL SETS FOR EIGENFUNCTIONS OF THE LAPLACIAN
ON THE TORUS
Zeev Rudnick and Igor Wigman
We study the volume of nodal sets for eigenfunctions of the Laplacian
on the
standard torus in two or more dimensions. We consider a sequence of
eigenvalues
$4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and
compute the
expectation and variance of the volume of the nodal set with respect
to a
Gaussian probability measure on the eigenspaces. We show that the
expected
volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main
result is that
the variance of the volume normalized by $\sqrt{\eigenvalue}$ is
bounded by
$O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing
fluctuations
as we increase the dimension of the eigenspace.
http://front.math.ucdavis.edu/math-ph/0702081
---------------------------------------------------------------
5293. A PORTFOLIO DECOMPOSITION FORMULA
Traian A Pirvu and Ulrich G Haussmann
This paper derives a portfolio decomposition formula when the agent
maximizes
utility of her wealth at some finite planning horizon. The financial
market is
complete and consists of multiple risky assets (stocks) plus a risk
free asset.
The stocks are modelled as exponential Brownian motions with drift and
volatility being Ito processes. The optimal portfolio has two
components: a
myopic component and a hedging one. We show that the myopic component
is robust
with respect to stopping times. We employ the Clark-Haussmann formula
to derive
portfolio s hedging component.
http://front.math.ucdavis.edu/math.PR/0702726
---------------------------------------------------------------
5294. ON ROBUST UTILITY MAXIMIZATION
Traian A Pirvu and Ulrich G Haussmann
This paper studies the problem of optimal investment in incomplete
markets,
robust with respect to stopping times. We work on a Brownian motion
framework
and the stopping times are adapted to the Brownian filtration.
Robustness can
only be achieved for logartihmic utility, otherwise a cashflow should
be added
to the investor s wealth. The cashflow can be decomposed into the sum
of an
increasing and a decreasing process. The last one can be viewed as
consumption.
The first one is an insurance premium the agent has to pay.
http://front.math.ucdavis.edu/math.PR/0702727
---------------------------------------------------------------
5295. MATRIX NORMS AND RAPID MIXING FOR SPIN SYSTEMS
Martin Dyer and Leslie Ann Goldberg and Mark Jerrum
We give a systematic development of the application of matrix norms
to rapid
mixing in spin systems. We show that rapid mixing of both random
update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix
norm of the
associated dependency matrix is less than 1. We give improved
analysis for the
case in which the diagonal of the dependency matrix is 0 (as in heat
bath
dynamics). We apply the matrix norm methods to random update and
systematic
scan Glauber dynamics for colouring various classes of graphs. We give a
general method for estimating a norm of a symmetric non-regular
matrix. This
leads to improved mixing times for any class of graphs which is
hereditary and
sufficiently sparse including several classes of degree-bounded
graphs such as
non-regular graphs, trees, planar graphs and graphs with given tree-
width and
genus.
http://front.math.ucdavis.edu/math.PR/0702744
---------------------------------------------------------------
5296. ASYMPTOTICS OF THE MINIMUM MANIPULATING COALITION SIZE FOR
POSITIONAL VOTING RULES UNDER IC BEHAVIOUR
Geoffrey Pritchard and Mark C. Wilson
We consider the problem of manipulation of elections using positional
voting
rules under Impartial Culture voter behaviour. We consider both the
logical
possibility of coalitional manipulation, and the number of voters
that must be
recruited to form a manipulating coalition. It is shown that the
manipulation
problem may be well approximated by a very simple linear program in two
variables. This permits a comparative analysis of the asymptotic
(large-population) manipulability of the various rules. It is seen
that the
manipulation resistance of positional rules with 5 or 6 (or more)
candidates is
quite different from the more commonly analyzed 3- and 4-candidate
cases.
http://front.math.ucdavis.edu/math.PR/0702752
---------------------------------------------------------------
5297. CLASSICAL DILATIONS \`A LA HUDSON-PARTHASARATHY OF MARKOV
SEMIGROUPS
M. Gregoratti
We study the Classical Probability analogue of the dilations of a
quantum
dynamical semigroup defined in Quantum Probability via quantum
stochastic
differential equations. Given a homogeneous Markov chain in
continuous time in
a finite state space E, we introduce a second system, an environment,
and a
deterministic invertible time-homogeneous global evolution of the
system E with
this environment such that the original Markov evolution of E can be
realized
by a proper choice of the initial random state of the environment. We
also
compare this dilations with the dilations of a quantum dynamical
semigroup in
Quantum Probability: given a classical Markov semigroup, we extend it
to a
proper quantum dynamical semigroup for which we can find a Hudson-
Parthasarathy
dilation which is itself an extension of our classical dilation.
http://front.math.ucdavis.edu/math.PR/0702784
---------------------------------------------------------------
5298. FURTHER RESULTS ON SOME SINGULAR LINEAR STOCHASTIC
DIFFERENTIAL EQUATIONS
Larbi Alili and Ching-Tang Wu
A class of Volterra transforms, preserving the Wiener measure, with
kernels
of Goursat type is considered. We provide some results on the
inverses of the
associated Gramian matrices. These are applied to the study of a
class of
linear singular stochastic differential equations together with the
corresponding decompositions of filtrations. The studied equations
are viewed
as non-canonical decompositions of some generalized bridges.
http://front.math.ucdavis.edu/math.PR/0702785
---------------------------------------------------------------
5299. STOCHASTIC HAMILTONIAN DYNAMICAL SYSTEMS
Joan-Andreu L\'azaro-Cam\'{\i} and Juan-Pablo Ortega
We use the global stochastic analysis tools introduced by P. A. Meyer
and L.
Schwartz to write down a stochastic generalization of the Hamilton
equations on
a Poisson manifold that, for exact symplectic manifolds, satisfy a
natural
critical action principle similar to the one encountered in classical
mechanics. Several features and examples in relation with the solution
semimartingales of these equations are presented.
http://front.math.ucdavis.edu/math.PR/0702787
---------------------------------------------------------------
5300. MINIMAL POSITION AND CRITICAL MARTINGALE CONVERGENCE IN
BRANCHING RANDOM WALKS, AND DIRECTED POLYMERS ON DISORDERED TREES
Yueyun Hu (LAGA) and Zhan Shi (PMA)
We establish a second-order almost sure limit theorem for the minimal
position in a one-dimensional super-critical branching random walk,
and also
prove a martingale convergence theorem which answers a question of
Biggins and
Kyprianou (2005). Our method applies furthermore to the study of
directed
polymers on a disordered tree; in particular, we give a rigorous
proof of a
phase transition phenomenon for the partition function, described by
Derrida
and Spohn (1988).
http://front.math.ucdavis.edu/math.PR/0702799
---------------------------------------------------------------
5301. ORBITAL APPROACH TO MICROSTATE FREE ENTROPY
Fumio Hiai and Takuho Miyamoto and Yoshimichi Ueda
Motivated by Voiculescu's liberation theory, we introduce the orbital
free
entropy $\chi_orb$ for non-commutative self-adjoint random variables
(also for
"hyperfinite random multivariables"). Besides its basic properties
the relation
of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the
dimension counterpart of $\chi_orb$ is discussed.
http://front.math.ucdavis.edu/math.OA/0702745
---------------------------------------------------------------
5302. POISSON PROCESS APPROXIMATION: FROM PALM THEORY TO STEIN'S METHOD
Louis H. Y. Chen and Aihua Xia
This exposition explains the basic ideas of Stein's method for
Poisson random
variable approximation and Poisson process approximation from the
point of view
of the immigration-death process and Palm theory. The latter approach
also
enables us to define local dependence of point processes [Chen and
Xia (2004)]
and use it to study Poisson process approximation for locally
dependent point
processes and for dependent superposition of point processes.
http://front.math.ucdavis.edu/math.PR/0702820
---------------------------------------------------------------
5303. PRICE SYSTEMS FOR MARKETS WITH TRANSACTION COSTS AND CONTROL
PROBLEMS FOR SOME FINANCE PROBLEMS
Tzuu-Shuh Chiang and Shang-Yuan Shiu and Shuenn-Jyi Sheu
In a market with transaction costs, the price of a derivative can be
expressed in terms of (preconsistent) price systems (after Kusuoka
(1995)). In
this paper, we consider a market with binomial model for stock price and
discuss how to generate the price systems. From this, the price
formula of a
derivative can be reformulated as a stochastic control problem. Then the
dynamic programming approach can be used to calculate the price. We also
discuss optimization of expected utility using price systems.
http://front.math.ucdavis.edu/math.PR/0702828
---------------------------------------------------------------
5304. ASYMPTOTIC ARBITRAGE AND NUM\'ERAIRE PORTFOLIOS IN LARGE
FINANCIAL MARKETS
Dmitry B. Rokhlin
This paper deals with the notion of a large financial market and the
concepts
of asymptotic arbitrage and strong asymptotic arbitrage (both of the
first
kind), introduced by Yu.M. Kabanov and D.O. Kramkov. We show that the
arbitrage
properties of a large market are completely determined by the asymptotic
behavior of the sequence of the num\'eraire portfolios, related to
the small
markets. The obtained criteria can be expressed in terms of
contiguity, entire
separation and Hellinger integrals, provided these notions are
extended to
sub-probability measures. As examples we consider market models on
finite
probability spaces, semimartingale and diffusion models. Also a
discrete-time
infinite horizon market model with one log-normal stock is examined.
http://front.math.ucdavis.edu/math.PR/0702849
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