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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.
Author(s): S. Rezakhah and S. Shemehsavar
Abstract: 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://arXiv.org/abs/math/0701019
http://front.math.ucdavis.edu/math.PR/0701019
(alternate)
Author(s): Lancelot F. James and Marc Yor
Abstract: 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://arXiv.org/abs/math/0701049
http://front.math.ucdavis.edu/math.PR/0701049
(alternate)
Author(s): N. Serdyukova
Abstract: 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://arXiv.org/abs/math/0701058
http://front.math.ucdavis.edu/math.PR/0701058
(alternate)
Author(s): Ilia Negri and Yoichi Nishiyama
Abstract: 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://arXiv.org/abs/math/0701022
http://front.math.ucdavis.edu/math.ST/0701022
(alternate)
Author(s): Matteo Ortisi (Dept. of Mathematics and University of Milano)
Abstract: 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://arXiv.org/abs/math/0701095
http://front.math.ucdavis.edu/math.PR/0701095
(alternate)
Author(s): Holger Rauhut and Karin Schnass and Pierre Vandergheynst
Abstract: 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://arXiv.org/abs/math/0701131
http://front.math.ucdavis.edu/math.PR/0701131
(alternate)
Author(s): David J. Aldous and Wilfrid S. Kendall
Abstract: 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://arXiv.org/abs/math/0701140
http://front.math.ucdavis.edu/math.PR/0701140
(alternate)
Author(s): Yu Miao and Guangyu Yang and Luming Shen
Abstract: 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://arXiv.org/abs/math/0701162
http://front.math.ucdavis.edu/math.PR/0701162
(alternate)
Author(s): Yu Miao and Guangyu Yang
Abstract: 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://arXiv.org/abs/math/0701167
http://front.math.ucdavis.edu/math.PR/0701167
(alternate)
Author(s): Lincoln Chayes and Thomas M. Liggett
Abstract: 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://arXiv.org/abs/math/0701180
http://front.math.ucdavis.edu/math.PR/0701180
(alternate)
Author(s): Yu Miao and Guangyu Yang
Abstract: 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://arXiv.org/abs/math/0701183
http://front.math.ucdavis.edu/math.PR/0701183
(alternate)
Author(s): Witold Bednorz
Abstract: 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://arXiv.org/abs/math/0701191
http://front.math.ucdavis.edu/math.PR/0701191
(alternate)
Author(s): Itai Benjamini and Ariel Yadin
Abstract: 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://arXiv.org/abs/math/0701201
http://front.math.ucdavis.edu/math.PR/0701201
(alternate)
Author(s): Antoine Lejay
Abstract: 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://arXiv.org/abs/math/0701219
http://front.math.ucdavis.edu/math.PR/0701219
(alternate)
Author(s): Iain MacPhee and Mikhail Menshikov and Dimitri Petritis and and Serguei Popov
Abstract: 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://arXiv.org/abs/math/0701226
http://front.math.ucdavis.edu/math.PR/0701226
(alternate)
Author(s): Marek Biskup and Timothy M. Prescott
Abstract: 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://arXiv.org/abs/math/0701248
http://front.math.ucdavis.edu/math.PR/0701248
(alternate)
Author(s): Pietro Caputo and Alessandra Faggionato
Abstract: 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://arXiv.org/abs/math/0701253
http://front.math.ucdavis.edu/math.PR/0701253
(alternate)
Author(s): James Allen Fill and Svante Janson
Abstract: 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://arXiv.org/abs/math/0701259
http://front.math.ucdavis.edu/math.PR/0701259
(alternate)
Author(s): Sonia Fourati
Abstract: 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://arXiv.org/abs/math/0701271
http://front.math.ucdavis.edu/math.PR/0701271
(alternate)
Author(s): Evelina Shamarova
Abstract: 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://arXiv.org/abs/math/0701281
http://front.math.ucdavis.edu/math.PR/0701281
(alternate)
Author(s): Pierre Del Moral (JAD and IRISA / INRIA Rennes) and Laurent Miclo (LATP) and Fr\'{e}d\'{e}ric Patras (JAD), Sylvain Rubenthaler (JAD)
Abstract: 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://arXiv.org/abs/math/0701284
http://front.math.ucdavis.edu/math.PR/0701284
(alternate)
Author(s): Svante Janson
Abstract: 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://arXiv.org/abs/math/0701288
http://front.math.ucdavis.edu/math.PR/0701288
(alternate)
Author(s): Asaf Nachmias and Yuval Peres
Abstract: 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://arXiv.org/abs/math/0701316
http://front.math.ucdavis.edu/math.PR/0701316
(alternate)
Author(s): Wlodek Bryc
Abstract: 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://arXiv.org/abs/math/0701318
http://front.math.ucdavis.edu/math.PR/0701318
(alternate)
Author(s): Philipp Pluch
Abstract: 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://arXiv.org/abs/math/0701323
http://front.math.ucdavis.edu/math.ST/0701323
(alternate)
Author(s): Satya N. Majumdar
Abstract: 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://arXiv.org/abs/cond-mat/0701193
http://front.math.ucdavis.edu/cond-mat/0701193
(alternate)
Author(s): B. Bollobas and C. Borgs and J. Chayes and O. Riordan
Abstract: 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://arXiv.org/abs/math/0701346
http://front.math.ucdavis.edu/math.PR/0701346
(alternate)
Author(s): Pavel Chebotarev
Abstract: 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://arXiv.org/abs/math/0701359
http://front.math.ucdavis.edu/math.PR/0701359
(alternate)
Author(s): Philippe Chassaing (IECN) and Lucas Gerin (IECN)
Abstract: 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://arXiv.org/abs/math/0701347
http://front.math.ucdavis.edu/math.ST/0701347
(alternate)
Author(s): Evelina Shamarova
Abstract: 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://arXiv.org/abs/math/0701360
http://front.math.ucdavis.edu/math.PR/0701360
(alternate)
Author(s): Charles Bordenave and David McDonald and Alexandre Proutiere
Abstract: 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://arXiv.org/abs/math/0701363
http://front.math.ucdavis.edu/math.PR/0701363
(alternate)
Author(s): Kazumasa Kuwada
Abstract: 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://arXiv.org/abs/math/0701372
http://front.math.ucdavis.edu/math.PR/0701372
(alternate)
Author(s): Itai Benjamini and Ben Morris
Abstract: 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://arXiv.org/abs/math/0701390
http://front.math.ucdavis.edu/math.PR/0701390
(alternate)
Author(s): Johan Segers
Abstract: 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://arXiv.org/abs/math/0701411
http://front.math.ucdavis.edu/math.PR/0701411
(alternate)
Author(s): Glauco Valle
Abstract: 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://arXiv.org/abs/math/0701413
http://front.math.ucdavis.edu/math.PR/0701413
(alternate)
Author(s): Amir Dembo and Alain-Sol Sznitman
Abstract: 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://arXiv.org/abs/math/0701414
http://front.math.ucdavis.edu/math.PR/0701414
(alternate)
Author(s): Pablo A. Ferrari and James B. Martin and Leandro P. R. Pimentel
Abstract: 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://arXiv.org/abs/math/0701418
http://front.math.ucdavis.edu/math.PR/0701418
(alternate)
Author(s): Marc Lelarge
Abstract: 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://arXiv.org/abs/math/0701420
http://front.math.ucdavis.edu/math.PR/0701420
(alternate)
Author(s): Francesco Mainardi and Rudolf Gorenflo and Enrico Scalas
Abstract: 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://arXiv.org/abs/math/0701454
http://front.math.ucdavis.edu/math.PR/0701454
(alternate)
Author(s): Francesco Mainardi and Rudolf Gorenflo and Alessandro Vivoli
Abstract: 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://arXiv.org/abs/math/0701455
http://front.math.ucdavis.edu/math.PR/0701455
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: 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 $t$, $L^{lower}L^{upper}\}$ and
$q_i$=$\lim_{t\to\infty}\mathbf{P}\{L_t=i\}$ ($L^{lower}
http://arXiv.org/abs/math/0701458
http://front.math.ucdavis.edu/math.PR/0701458
(alternate)
Author(s): Sourav Chatterjee and Elizabeth Meckes
Abstract: 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://arXiv.org/abs/math/0701464
http://front.math.ucdavis.edu/math.PR/0701464
(alternate)
Author(s): A. Guionnet and D. Shlyakhtenko
Abstract: 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://arXiv.org/abs/math/0701465
http://front.math.ucdavis.edu/math.PR/0701465
(alternate)
Author(s): Elchanan Mossel and Dror Weitz and Nicholas Wormald
Abstract: 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://arXiv.org/abs/math/0701471
http://front.math.ucdavis.edu/math.PR/0701471
(alternate)
Author(s): Nikolaos Fountoulakis and Bruce Reed
Abstract: 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://arXiv.org/abs/math/0701474
http://front.math.ucdavis.edu/math.CO/0701474
(alternate)
Author(s): Laurent Mazliak (PMA and IMJ)
Abstract: 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://arXiv.org/abs/math/0701490
http://front.math.ucdavis.edu/math.HO/0701490
(alternate)
Author(s): Chandra Nair and Prasad Tetali
Abstract: 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://arXiv.org/abs/math/0701494
http://front.math.ucdavis.edu/math.PR/0701494
(alternate)
Author(s): S. Geiss and S. Montgomery-Smith and E. Saksman
Abstract: 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://arXiv.org/abs/math/0701516
http://front.math.ucdavis.edu/math.CA/0701516
(alternate)
Author(s): Joseph Najnudel
Abstract: 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://arXiv.org/abs/math/0701526
http://front.math.ucdavis.edu/math.PR/0701526
(alternate)
Author(s): Anja Sturm and Jan Swart
Abstract: 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://arXiv.org/abs/math/0701555
http://front.math.ucdavis.edu/math.PR/0701555
(alternate)
Author(s): Fabio Scarabotti and Filippo Tolli
Abstract: 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://arXiv.org/abs/math/0701533
http://front.math.ucdavis.edu/math.RT/0701533
(alternate)
Author(s): J. Davidsen and P. Grassberger and M. Paczuski
Abstract: 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://arXiv.org/abs/physics/0701190
http://front.math.ucdavis.edu/physics/0701190
(alternate)
Author(s): Yuri Bakhtin
Abstract: 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://arXiv.org/abs/math/0701569
http://front.math.ucdavis.edu/math.PR/0701569
(alternate)
Author(s): Martin Hildebrand and Joseph McCollum
Abstract: 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://arXiv.org/abs/math/0701570
http://front.math.ucdavis.edu/math.PR/0701570
(alternate)
Author(s): Boris L. Granovsky and Dudley Stark and Michael Erlihson
Abstract: 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://arXiv.org/abs/math/0701584
http://front.math.ucdavis.edu/math.PR/0701584
(alternate)
Author(s): Fabio Scarabotti and Filippo Tolli
Abstract: 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://arXiv.org/abs/math/0701603
http://front.math.ucdavis.edu/math.PR/0701603
(alternate)
Author(s): Maxim Krikun (IECN)
Abstract: 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://arXiv.org/abs/math/0701611
http://front.math.ucdavis.edu/math.PR/0701611
(alternate)
Author(s): Catherine Greenhill and Brendan D. McKay
Abstract: 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://arXiv.org/abs/math/0701600
http://front.math.ucdavis.edu/math.CO/0701600
(alternate)
Author(s): Alexey M. Kulik
Abstract: 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://arXiv.org/abs/math/0701632
http://front.math.ucdavis.edu/math.PR/0701632
(alternate)
Author(s): K.B. Athreya and A.P. Ghosh and S. Sethuraman
Abstract: 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://arXiv.org/abs/math/0701649
http://front.math.ucdavis.edu/math.PR/0701649
(alternate)
Author(s): Thomas Simon (DP)
Abstract: 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://arXiv.org/abs/math/0701653
http://front.math.ucdavis.edu/math.PR/0701653
(alternate)
Author(s): Damien Pitman
Abstract: 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 01, 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://arXiv.org/abs/math/0701656
http://front.math.ucdavis.edu/math.PR/0701656
(alternate)
Author(s): A. J. Roberts
Abstract: 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://arXiv.org/abs/math/0701623
http://front.math.ucdavis.edu/math.DS/0701623
(alternate)
Author(s): Steven N. Evans and Tye Lidman
Abstract: 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://arXiv.org/abs/math/0701657
http://front.math.ucdavis.edu/math.PR/0701657
(alternate)
Author(s): Sunder Sethuraman
Abstract: 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://arXiv.org/abs/math/0701660
http://front.math.ucdavis.edu/math.PR/0701660
(alternate)
Author(s): Krishna Athreya and Siva Athreya and and Srikanth Iyer
Abstract: 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://arXiv.org/abs/math/0701661
http://front.math.ucdavis.edu/math.PR/0701661
(alternate)
Author(s): Yu Zhang
Abstract: 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://arXiv.org/abs/math/0701689
http://front.math.ucdavis.edu/math.PR/0701689
(alternate)
Author(s): Steven P. Lalley
Abstract: 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://arXiv.org/abs/math/0701698
http://front.math.ucdavis.edu/math.PR/0701698
(alternate)
Author(s): Alexander Gnedin and Ben Hansen and Jim Pitman
Abstract: 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://arXiv.org/abs/math/0701718
http://front.math.ucdavis.edu/math.PR/0701718
(alternate)
Author(s): Dominic Schuhmacher
Abstract: 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://arXiv.org/abs/math/0701728
http://front.math.ucdavis.edu/math.PR/0701728
(alternate)
Author(s): Y. Kondratiev and E. Lytvynov and M. R\"ockner
Abstract: 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://arXiv.org/abs/math/0701736
http://front.math.ucdavis.edu/math.PR/0701736
(alternate)
Author(s): Robin Pemantle
Abstract: 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://arXiv.org/abs/math/0701741
http://front.math.ucdavis.edu/math.PR/0701741
(alternate)
Author(s): Alexey M.Kulik
Abstract: 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://arXiv.org/abs/math/0701747
http://front.math.ucdavis.edu/math.PR/0701747
(alternate)
Author(s): G. Maillard and T. Mountford
Abstract: 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://arXiv.org/abs/math/0701754
http://front.math.ucdavis.edu/math.PR/0701754
(alternate)
Author(s): Christoph Richard and Iwan Jensen and Anthony J. Guttmann
Abstract: 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://arXiv.org/abs/math/0701633
http://front.math.ucdavis.edu/math.CO/0701633
(alternate)
Author(s): Robert S. Maier
Abstract: 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://arXiv.org/abs/q-bio/0701036
http://front.math.ucdavis.edu/q-bio.BM/0701036
(alternate)
Author(s): Karlheinz Groechenig and Benedikt M. Poetscher and Holger Rauhut
Abstract: 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://arXiv.org/abs/math/0701781
http://front.math.ucdavis.edu/math.PR/0701781
(alternate)
Author(s): Zach Dietz and Sunder Sethuraman
Abstract: 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://arXiv.org/abs/math/0701798
http://front.math.ucdavis.edu/math.PR/0701798
(alternate)
Author(s): M\'arton Isp\'any and Gyula Pap
Abstract: 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://arXiv.org/abs/math/0701803
http://front.math.ucdavis.edu/math.PR/0701803
(alternate)
Author(s): Guan-Yu Chen and Laurent Saloff-Coste
Abstract: 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://arXiv.org/abs/math/0701827
http://front.math.ucdavis.edu/math.PR/0701827
(alternate)
Author(s): B. D'Auria
Abstract: 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://arXiv.org/abs/math/0701842
http://front.math.ucdavis.edu/math.PR/0701842
(alternate)
Author(s): Philippe Briand (IRMAR) and Fulvia Confortola
Abstract: 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://arXiv.org/abs/math/0701849
http://front.math.ucdavis.edu/math.PR/0701849
(alternate)
Author(s): Iosif Pinelis
Abstract: 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://arXiv.org/abs/math/0701806
http://front.math.ucdavis.edu/math.ST/0701806
(alternate)
Author(s): Florencia G. Leonardi
Abstract: 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://arXiv.org/abs/math/0701810
http://front.math.ucdavis.edu/math.ST/0701810
(alternate)
Author(s): Frederic Dambreville (DGA/CTA/DT/GIP)
Abstract: 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://arXiv.org/abs/math/0701801
http://front.math.ucdavis.edu/math.LO/0701801
(alternate)
Author(s): A. Guionnet and D. Shlyakhtenko
Abstract: 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://arXiv.org/abs/math/0701787
http://front.math.ucdavis.edu/math.OA/0701787
(alternate)
Author(s): Stefan Ankirchner and Peter Imkeller and Goncalo Reis
Abstract: 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://arXiv.org/abs/math/0701875
http://front.math.ucdavis.edu/math.PR/0701875
(alternate)
Author(s): Benjamin Jourdain (CERMICS) and Florent Malrieu (IRMAR)
Abstract: 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://arXiv.org/abs/math/0701879
http://front.math.ucdavis.edu/math.PR/0701879
(alternate)
Author(s): Jonas H\"agg
Abstract: 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://arXiv.org/abs/math/0701880
http://front.math.ucdavis.edu/math.PR/0701880
(alternate)
Author(s): Yann Ollivier
Abstract: 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://arXiv.org/abs/math/0701886
http://front.math.ucdavis.edu/math.PR/0701886
(alternate)
Author(s): Celine Jost
Abstract: 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://arXiv.org/abs/math/0701888
http://front.math.ucdavis.edu/math.PR/0701888
(alternate)
Author(s): Catherine Greenhill (University of New South Wales) and Fred B. Holt (University of Washington), Nicholas Wormald (University of Waterloo)
Abstract: 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://arXiv.org/abs/math/0701863
http://front.math.ucdavis.edu/math.CO/0701863
(alternate)
Author(s): Robert C. Griffiths and Dario Span\'{o}
Abstract: 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://arXiv.org/abs/math/0701897
http://front.math.ucdavis.edu/math.PR/0701897
(alternate)
Author(s): S\'{e}bastien Darses (PMA) and Ivan Nourdin (PMA) and Giovanni Peccati (LSTA)
Abstract: 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://arXiv.org/abs/math/0701910
http://front.math.ucdavis.edu/math.PR/0701910
(alternate)
Author(s): Vladimir Vatutin and Vitali Wachtel
Abstract: 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://arXiv.org/abs/math/0701914
http://front.math.ucdavis.edu/math.PR/0701914
(alternate)
Author(s): Vincent Bansaye (PMA)
Abstract: 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://arXiv.org/abs/math/0701917
http://front.math.ucdavis.edu/math.PR/0701917
(alternate)
Author(s): Denis Denisov and Serguei Foss and Dmitry Korshunov
Abstract: 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://arXiv.org/abs/math/0701920
http://front.math.ucdavis.edu/math.PR/0701920
(alternate)
Author(s): Tetyana Kadankova and No\"{e}l Veraverbeke
Abstract: 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://arXiv.org/abs/math/0701924
http://front.math.ucdavis.edu/math.PR/0701924
(alternate)
Author(s): Thomas Hofmann and Bernhard Sch\"olkopf and Alexander J. Smola
Abstract: 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://arXiv.org/abs/math/0701907
http://front.math.ucdavis.edu/math.ST/0701907
(alternate)
Author(s): Morris L. Eaton and James P. Hobert and Galin L. Jones and Wen-Lin Lai
Abstract: 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://arXiv.org/abs/math/0701938
http://front.math.ucdavis.edu/math.ST/0701938
(alternate)
Author(s): Dmitry Panchenko
Abstract: 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://arXiv.org/abs/math-ph/0701074
http://front.math.ucdavis.edu/math-ph/0701074
(alternate)
Author(s): Evi Daems and Arno Kuijlaars and and Wim Veys
Abstract: 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://arXiv.org/abs/math/0701923
http://front.math.ucdavis.edu/math.CV/0701923
(alternate)
Author(s): Daniel Levin and Mark Wildon
Abstract: 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://arXiv.org/abs/math/0702002
http://front.math.ucdavis.edu/math.PR/0702002
(alternate)
Author(s): Amir Dembo and Andrea Montanari
Abstract: 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 rho0. 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://arXiv.org/abs/math/0702007
http://front.math.ucdavis.edu/math.PR/0702007
(alternate)
Author(s): Andrew D. Barbour and Vydas Cekanavicius and Aihua Xia
Abstract: 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://arXiv.org/abs/math/0702008
http://front.math.ucdavis.edu/math.PR/0702008
(alternate)
Author(s): Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (BR-IMPA) and Maria Eulalia Vares (BR-CBPF)
Abstract: 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://arXiv.org/abs/math/0702012
http://front.math.ucdavis.edu/math.PR/0702012
(alternate)
Author(s): Matthias Birkner and Iljana Z\"ahle
Abstract: 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://arXiv.org/abs/math/0702020
http://front.math.ucdavis.edu/math.PR/0702020
(alternate)
Author(s): Tom Britton and Svante Janson and Anders Martin-Lof
Abstract: 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 th | |
