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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 that infected individuals infect their not yet infected
friends independently with probability p. For this situation the paper
determines R_0 and tau_0, the basic reproduction number and the asymptotic
final size in case of a major outbreak. Further, the paper looks at some
different local vaccination strategies where individuals are chosen randomly
and vaccinated, or friends of the selected individuals are vaccinated, prior to
the introduction of the disease. For the studied vaccination strategies the
paper determines R_v: the reproduction number, and tau_v: the asymptotic final
proportion infected in case of a major outbreak, after vaccinating a fraction
v.
http://arXiv.org/abs/math/0702021
http://front.math.ucdavis.edu/math.PR/0702021
(alternate)
Author(s): Sandrine Peche and Alexander Soshnikov
Abstract: We show that the spectral radius of an $N\times N$ random symmetric matrix
with i.i.d. bounded centered but non-symmetrically distributed entries is
bounded from above by $ 2 \*\sigma + o(N^{-6/11+\epsilon}), $ where $\sigma^2 $
is the variance of the matrix entries and $\epsilon $ is an arbitrary small
positive number. Our bound improves the earlier results by Z.F\"{u}redi and
J.Koml\'{o}s (1981), and the recent bound obtained by Van Vu (2005).
http://arXiv.org/abs/math/0702035
http://front.math.ucdavis.edu/math.PR/0702035
(alternate)
Author(s): Christian Houdr\'e and Heinrich Matzinger
Abstract: We investigate the variance of the optimal alignment score of two independent
iid binary, with parameter 1/2, sequences of length $n$. The scoring function
is such that one letter has a somewhat larger score than the other letter. In
this setting, we prove that the variance is of order $n$, and this confirms
Waterman's conjecture in this case.
http://arXiv.org/abs/math/0702036
http://front.math.ucdavis.edu/math.PR/0702036
(alternate)
Author(s): Marta Sanz-Sol\'e and Iv\'an Torrecilla-Tarantino
Abstract: For a fractional Brownian motion $B^H$ with Hurst parameter
$H\in]{1/4},{1/2}[\cup]{1/2},1[$, multiple indefinite integrals on a simplex
are constructed and the regularity of their sample paths are studied. Then, it
is proved that the family of probability laws of the processes obtained by
replacing $B^H$ by $\epsilon^{{1/2}} B^H$ satisfies a large deviation principle
in H\"older norm. The definition of the multiple integrals relies upon a
representation of the fractional Brownian motion in terms of a stochastic
integral with respect to a standard Brownian motion. For the large deviation
principle, the abstract general setting given by Ledoux in [Lecture Notes in
Math., vol. 1426 (1990) 1-14] is used.
http://arXiv.org/abs/math/0702049
http://front.math.ucdavis.edu/math.PR/0702049
(alternate)
Author(s): Hermine Bierm\'{e} (MAP5) and C\'{e}line Lacaux (IECN)
Abstract: We investigate the sample paths regularity of operator scaling alpha-stable
random fields. Such fields were introduced as anisotropic generalizations of
self-similar fields and satisfy a scaling property for a real matrix E. In the
case of harmonizable operator scaling random fields, the sample paths are
locally H\"{o}lderian and their H\"{o}lder regularity is characterized by the
eigen decomposition with respect to E. In particular, the directional
H\"{o}lder regularity may vary and is given by the eigenvalues of E. In the
case of moving average operator scaling random alpha-stable random fields, with
0
http://arXiv.org/abs/math/0702050
http://front.math.ucdavis.edu/math.PR/0702050
(alternate)
Author(s): Stefan Adams
Abstract: Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ on some
fixed time interval $[0,\beta]$ with symmetrised initial-terminal condition.
That is, for any $i$, the terminal location of the $i$-th motion is affixed to
the initial point of the $\sigma(i)$-th motion, where $\sigma$ is a uniformly
distributed random permutation of $1,...,N$.
In this paper, we describe the large-N behaviour of the empirical path
measure (the mean of the Dirac measures in the $N$ paths) when $
\Lambda\uparrow\mathbb{R}^d $ and $ N/|\Lambda|\to\rho $. The rate function is
given as a variational formula involving a certain entropy functional and a
Fenchel-Legendre transform.
Depending on the dimension and the density $ \rho $, there is phase
transition behaviour for the empirical path measure. For certain parameters
(high density, large time horizon) and dimensions $ d\ge 3 $ the empirical path
measure is not supported on all paths $ [0,\infty)\to\mathbb{R}^d $ which
contain a bridge path of any finite multiple of the time horizon $ [0,\beta] $.
For dimensions $ d=1,2 $, and for small densities and small time horizon $
[0,\beta] $ in dimensions $ d\ge 3$, the empirical path measure is supported on
those paths. In the first regime a finite fraction of the motions lives in
cycles of infinite length.
We outline that this transition leads to an empirical path measure
interpretation of {\it Bose-Einstein condensation}, known for systems of
Bosons.
http://arXiv.org/abs/math/0702053
http://front.math.ucdavis.edu/math.PR/0702053
(alternate)
Author(s): Nicola Cufaro Petroni
Abstract: We analyze the Levy processes produced by means of two interconnected classes
of non stable, infinitely divisible distribution: the Variance Gamma and the
Student laws. While the Variance Gamma family is closed under convolution, the
Student one is not: this makes its time evolution more complicated. We prove
that -- at least for one particular type of Student processes suggested by
recent empirical results, and for integral times -- the distribution of the
process is a mixture of other types of Student distributions, randomized by
means of a new probability distribution. The mixture is such that along the
time the asymptotic behavior of the probability density functions always
coincide with that of the generating Student law. We put forward the conjecture
that this can be a general feature of the Student processes. We finally analyze
the Ornstein--Uhlenbeck process driven by our Levy noises and show a few
simulation of it.
http://arXiv.org/abs/math/0702058
http://front.math.ucdavis.edu/math.PR/0702058
(alternate)
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan
Abstract: Given $\omega\ge 1$, let $Z^2_{(\omega)}$ be the graph with vertex set $Z^2$
in which two vertices are joined if they agree in one coordinate and differ by
at most $\omega$ in the other. (Thus $Z^2_{(1)}$ is precisely $Z^2$.) Let
$p_c(\omega)$ be the critical probability for site percolation in
$Z^2_{(\omega)}$. Extending recent results of Frieze, Kleinberg, Ravi and
Debany, we show that $\lim_{\omega\to\infty} \omega\pc(\omega)=\log(3/2)$. We
also prove analogues of this result on the $n$-by-$n$ grid and in higher
dimensions, the latter involving interesting connections to Gilbert's continuum
percolation model. To prove our results, we explore the component of the origin
in a certain non-standard way, and show that this exploration is well
approximated by a certain branching process.
http://arXiv.org/abs/math/0702061
http://front.math.ucdavis.edu/math.PR/0702061
(alternate)
Author(s): E. Baake and R. Bialowons
Abstract: We consider two versions of stochastic population models with mutation and
selection. The first approach relies on a multitype branching process; here,
individuals reproduce and change type (i.e., mutate) independently of each
other, without restriction on population size. We analyze the equilibrium
behaviour of this model, both in the forward and in the backward direction of
time; the backward point of view emerges if the ancestry of individuals chosen
randomly from the present population is traced back into the past.
The second approach is the Moran model with selection. Here, the population
has constant size N. Individuals reproduce (at rates depending on their types),
the offspring inherits the parent's type, and replaces a randomly chosen
individual (to keep population size constant). Independently of the
reproduction process, individuals can change type. As in the branching model,
we consider the ancestral lines of single individuals chosen from the
equilibrium population. We use analytical results of Fearnhead (2002) to
determine the explicit properties, and parameter dependence, of the ancestral
distribution of types, and its relationship with the stationary distribution in
forward time.
http://arXiv.org/abs/q-bio/0702002
http://front.math.ucdavis.edu/q-bio.PE/0702002
(alternate)
Author(s): Tapan Mukhopadhyay and D.N. Basu
Abstract: Binding energy of isospin asymmetric nuclei can be accessed with minimally
modified formula along the lines of the liquid droplet model by partitioning
the symmetry term into volume and surface terms. The volume symmetry energy
coefficient extracted from finite nuclei provides a constraint on the nuclear
symmetry energy. This approach also yields the neutron skin of a finite nucleus
through its relationship with the volume and surface symmetry terms and the
Coulomb energy coefficient. The symmetry energy at saturation density obtained
from the isoscalar as well as isovector components of the density dependent M3Y
effective interaction is found to be in close agreement with the volume
symmetry energy coefficient extracted from the measured atomic masses.
http://arXiv.org/abs/nucl-th/0605001
http://front.math.ucdavis.edu/nucl-th/0605001
(alternate)
Author(s): Tom Kennedy
Abstract: We simulate several models of random curves in the half plane and numerically
compute their stochastic driving process (as given by the Loewner equation).
Our models include models whose scaling limit is the Schramm-Loewner evolution
(SLE) and models for which it is not. We study several tests of whether the
driving process is Brownian motion. We find that just testing the normality of
the process at a fixed time is not effective at determining if the process is
Brownian motion. Tests that involve the independence of the increments of
Brownian motion are much more effective. We also study the zipper algorithm for
numerically computing the driving function of a simple curve. We give an
implementation of this algorithm which runs in a time O(N^1.35) rather than the
usual O(N^2), where N is the number of points on the curve.
http://arXiv.org/abs/math/0702071
http://front.math.ucdavis.edu/math.PR/0702071
(alternate)
Author(s): Matyas Barczy and Alexander Bendikov and Gyula Pap
Abstract: We prove limit theorems for row sums of a rowwise independent infinitesimal
array of random variables with values in a locally compact Abelian group. First
we give a proof of Gaiser's theorem, since it does not have an easy access and
it is not complete. This theorem gives sufficient conditions for convergence of
the row sums, but the limit measure can not have a nondegenerate idempotent
factor. Then we prove necessary and sufficient conditions for convergence of
the row sums, where the limit measure can be also a nondegenerate Haar measure
on a compact subgroup. Finally, we investigate special cases: the torus group,
the group of p-adic integers and the p-adic solenoid.
http://arXiv.org/abs/math/0702078
http://front.math.ucdavis.edu/math.PR/0702078
(alternate)
Author(s): Youri Davydov (Universite de Lille 1) and Vladimir Rotar (San Diego State University)
Abstract: We consider the invariance principle without the classical condition of
asymptotic negligibility of individual terms. More precisely, we explore the
difference of the following two distributions in the space C (of continuous
functions on [0,1]). The first is the distribution of the continuous piecewise
linear partial-sum process generated by a sequence of independent random
variables, and the second is the distribution of the similar process generated
by the sequence of normal r.v.'s with the same first two moments. The novelty
is that the condition of negligibility of the r.v.'s is not imposed. We
establish a necessary and sufficient condition of the weak convergence of the
difference mentioned to zero measure in C.
http://arXiv.org/abs/math/0702085
http://front.math.ucdavis.edu/math.PR/0702085
(alternate)
Author(s): Denis Bell
Abstract: Let $x$ denote a diffusion process defined on a closed compact manifold. In
an earlier article, the author introduced a new approach to constructing
admissible vector fields on the associated space of paths, under the assumption
of ellipticity of $x$. In this article, this method is extended to yield
similar results for degenerate diffusion processes. In particular, these
results apply to non-elliptic diffusions satisfying H\"ormander's condition.
http://arXiv.org/abs/math/0702092
http://front.math.ucdavis.edu/math.PR/0702092
(alternate)
Author(s): Jan M. Swart
Abstract: Certain Markov processes, or deterministic evolution equations, have the
property that they are dual to a stochastic process that exhibits extinction
versus unbounded growth, i.e., the total mass in such a process either becomes
zero, or grows without bounds as time tends to infinity. If this is the case,
then this phenomenon can often be used to determine the invariant measures, or
fixed points, of the process originally under consideration, and to study
convergence to equilibrium. This principle, which has been known since early
work on multitype branching processes, is here demonstrated on three new
examples with applications in the theory of interacting particle systems.
http://arXiv.org/abs/math/0702095
http://front.math.ucdavis.edu/math.PR/0702095
(alternate)
Author(s): Celine Jost
Abstract: We derive a class of ergodic transformations of self-similar Gaussian
processes that are Volterra, i.e. of type X_t = int^t_0 z_X(t,s)dW_s, t>0,
where z_X is a deterministic kernel and W is a standard Brownian motion.
http://arXiv.org/abs/math/0702096
http://front.math.ucdavis.edu/math.PR/0702096
(alternate)
Author(s): Dmitry Dolgopyat and Gerhard Keller and and Carlangelo Liverani
Abstract: We prove a quenched central limit theorem for random walks with bounded
increments in a randomly evolving environment on Zd. We assume that the
transition probabilities of the walk depend not too strongly on the environment
and that the evolution of the environment is Markovian with strong spatial and
temporal mixing properties.
http://arXiv.org/abs/math/0702100
http://front.math.ucdavis.edu/math.PR/0702100
(alternate)
Author(s): Henrik Hult and Gennady Samorodnitsky
Abstract: A random vector $X$ with representation $X = \sum_{j \geq 0} A_j Z_j$ is
considered. Here $(Z_j)$ is a sequence of independent and identically
distributed random vectors and $(A_j)$ is a sequence of random matrices,
``predictable'' with respect to the sequence $(Z_j)$. The distribution of $Z_1$
is assumed to be multivariate regular varying. Moment conditions on the
matrices $(A_j)$ are determined under which the distribution of $X$ is
regularly varying and, in fact, ``inherits'' its regular variation from that of
$(Z_j)$'s. We compute the associated limiting measure. Examples include linear
processes, random coefficient linear processes such as stochastic recurrence
equations, random sums, and stochastic integrals.
http://arXiv.org/abs/math/0702112
http://front.math.ucdavis.edu/math.PR/0702112
(alternate)
Author(s): T. Imamura and T. Sasamoto
Abstract: The one-dimensional totally asymmetric simple exclusion process (TASEP) is
considered. We study the time evolution property of a tagged particle in TASEP
with the step-type initial condition. Calculated is the multi-time joint
distribution function of its position. Using the relation of the dynamics of
TASEP to the Schur process, we show that the function is represented as the
Fredholm determinant. We also study the scaling limit. The universality of the
largest eigenvalue in the random matrix theory is realized in the limit. When
the hopping rates of all particles are the same, it is found that the joint
distribution function converges to that of the Airy process after the time at
whichthe particle begins to move. On the other hand, when there are several
particles with small hopping rate in front of a tagged particle, the limiting
process changes at a certain time from the Airy process to the process of the
largest eigenvalue in the Hermitian multi-matrix model with external sources.
http://arXiv.org/abs/math-ph/0702009
http://front.math.ucdavis.edu/math-ph/0702009
(alternate)
Author(s): Francesco Mainardi and Gianni Pagnini and Rudolf Gorenflo
Abstract: The Mellin transform is usually applied in probability theory to the product
of independent random variables. In recent times the machinery of the Mellin
transform has been adopted to describe the L\'evy stable distributions, and
more generally the probability distributions governed by generalized diffusion
equations of fractional order in space and/or in time. In these cases the
related stochastic processes are self-similar and are simply referred to as
fractional diffusion processes. We provide some integral formulas involving the
distributions of these processes that can be interpreted in terms of
subordination laws.
http://arXiv.org/abs/math/0702133
http://front.math.ucdavis.edu/math.PR/0702133
(alternate)
Author(s): Irina Kourkova (PMA)
Abstract: Recently, Bauke and Mertens conjectured that the local statistics of energies
in random spin systems with discrete spin space should, in most circumstances,
be the same as in the random energy model. We show that this conjecture holds
true as well for directed polymers in random environment. We also show that,
under certain conditions, this conjecture holds for directed polymers even if
energy levels that grow moderately with the volume of the system are
considered.
http://arXiv.org/abs/math/0702149
http://front.math.ucdavis.edu/math.PR/0702149
(alternate)
Author(s): I. Florescu and A. Myasnikov and A. Mahalanobis
Abstract: This paper presents a novel methodology to test the security of the
Diffie-Hellman public key exchange protocol. The security of many cryptographic
schemes rely on the hardness of this problem. We are presenting a purely
statistical test to compare this problem in different groups. We are using
groups included in the Zp group with p prime as a major example, however the
methods presented are not restricted to these groups. The presentation of the
results is primarily intended to introduce novel applications of statistical
methodologies to the area of mathematical cryptography. As such we will
emphasize the cryptographical aspects of the work more than the statistical
notions.
http://arXiv.org/abs/math/0702155
http://front.math.ucdavis.edu/math.ST/0702155
(alternate)
Author(s): A.M.G.Cox and D.Hobson and J.Ob\l\'oj
Abstract: We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+
F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$
a local martingale. The concrete nature of the representation makes the
inequality useful for a variety of applications. In this work, we use the
inequalities to derive constructions and optimality results of Vallois'
Skorokhod embeddings. We discuss their financial interpretation in the context
of robust pricing and hedging of options written on the local time. In the
final part of the paper we use the inequalities to solve a class of optimal
stopping problems of the form $\sup_\tau E[F(L_\tau)-\int_0^\tau
\beta(B_s)ds]$. The solution is given via a minimal solution to a system of
differential equations and thus resembles the maximality principle described by
Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the
techniques.
http://arXiv.org/abs/math/0702173
http://front.math.ucdavis.edu/math.PR/0702173
(alternate)
Author(s): Alexey M. Kulik
Abstract: We consider sequences of additive functionals of difference approximations
for uniformly non-degenerate multidimensional diffusions. The conditions are
given, sufficient for such a sequence to converge weakly to a W-functional of
the limiting process. The class of the W-functionals, that can be obtained as
the limiting ones, is completely described in the terms of the associated
W-measures, and coincides with the class of the functionals that are regular
w.r.t. the phase variable.
http://arXiv.org/abs/math/0702175
http://front.math.ucdavis.edu/math.PR/0702175
(alternate)
Author(s): Y.G. Kondratiev and O.V. Kutoviy and E.W. Lytvynov
Abstract: A Kawasaki dynamics in continuum is a dynamics of an infinite system of
interacting particles in $\mathbb R^d$ which randomly hop over the space. In
this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs
measure $\mu$ as invariant measure. We study a diffusive limit of such a
dynamics, derived through a scaling of both the jump rate and time. Under weak
assumptions on the potential of pair interaction, $\phi$, (in particular,
admitting a singularity of $\phi$ at zero), we prove that, on a set of smooth
local functions, the generator of the scaled dynamics converges to the
generator of an equilibrium diffusive dynamics of an infinite system of
interacting particles. If the set on which the generators converge is a core
for the diffusion generator, the latter result implies the weak convergence of
finite-dimensional distributions of the corresponding equilibrium processes. In
particular, if the potential $\phi$ is from $C_{\mathrm b}^3(\R^d)$ and
sufficiently quickly converges to zero at infinity, we conclude from a result
in [Choi {\it et al.}, {J. Math. Phys.} {39} (1998) 6509--6536] that the
convergence of processes holds when the limiting diffusion is the gradient
stochastic dynamics.
http://arXiv.org/abs/math/0702178
http://front.math.ucdavis.edu/math.PR/0702178
(alternate)
Author(s): Aernout C.D. van Enter and Tim Hulshof
Abstract: In this note we analyze an anisotropic, two-dimensional bootstrap percolation
model introduced by Gravner and Griffeath. We present upper and lower bounds on
the finite-size effects. We discuss the similarities with the semi-oriented
model introduced by Duarte.
http://arXiv.org/abs/cond-mat/0702145
http://front.math.ucdavis.edu/cond-mat/0702145
(alternate)
Author(s): M.T. Barlow and R.F. Bass and and T. Kumagai
Abstract: We investigate the relationships between the parabolic Harnack inequality,
heat kernel estimates, some geometric conditions, and some analytic conditions
for random walks with long range jumps. Unlike the case of diffusion processes,
the parabolic Harnack inequality does not, in general, imply the corresponding
heat kernel estimates.
http://arXiv.org/abs/math/0702221
http://front.math.ucdavis.edu/math.PR/0702221
(alternate)
Author(s): Oliver Riordan and Mark Walters
Abstract: We use the method of Balister, Bollobas and Walters to give rigorous 99.9999%
confidence intervals for the critical probabilities for site and bond
percolation on the 11 Archimedean lattices. In our computer calculations, the
emphasis is on simplicity and ease of verification, rather than obtaining the
best possible results. Nevertheless, we obtain intervals of width at most
0.0005 in all cases.
http://arXiv.org/abs/math/0702232
http://front.math.ucdavis.edu/math.PR/0702232
(alternate)
Author(s): Thomas Strohmer and Roman Vershynin
Abstract: The Kaczmarz method for solving linear systems of equations is an iterative
algorithm that has found many applications ranging from computer tomography to
digital signal processing. Despite the popularity of this method, useful
theoretical estimates for its rate of convergence are still scarce. We
introduce a randomized version of the Kaczmarz method for consistent,
overdetermined linear systems and we prove that it converges with expected
exponential rate. Furthermore, this is the first solver whose rate does not
depend on the number of equations in the system. The solver does not even need
to know the whole system, but only a small random part of it. It thus
outperforms all previously known methods on general extremely overdetermined
systems. Even for moderately overdetermined systems, numerical simulations as
well as theoretical analysis reveal that our algorithm can converge faster than
the celebrated conjugate gradient algorithm. Furthermore, our theory and
numerical simulations confirm a prediction of Feichtinger et al. in the context
of reconstructing bandlimited functions from nonuniform sampling.
http://arXiv.org/abs/math/0702226
http://front.math.ucdavis.edu/math.NA/0702226
(alternate)
Author(s): Fred Kochman and Jim Reeds
Abstract: According to a 1975 result of T. Kaijser, if some nonvanishing product of
hidden Markov model (HMM) stepping matrices is subrectangular, and the
underlying chain is aperiodic, the corresponding $\alpha$-chain has a unique
invariant limiting measure $\lambda$. Here the $\alpha$-chain
$\{\alpha_n\}=\{(\alpha_{ni})\}$ is given by \[\alpha_{ni}=P(X_n=i|
Y_n,Y_{n-1},...),\] where $\{(X_n,Y_n)\}$ is a finite state HMM with unobserved
Markov chain component $\{X_n\}$ and observed output component $\{Y_n\}$. This
defines $\{\alpha_n\}$ as a stochastic process taking values in the probability
simplex. It is not hard to see that $\{\alpha_n\}$ is itself a Markov chain.
The stepping matrices $M(y)=(M(y)_{ij})$ give the probability that
$(X_n,Y_n)=(j,y)$, conditional on $X_{n-1}=i$. A matrix is said to be
subrectangular if the locations of its nonzero entries forms a cartesian
product of a set of row indices and a set of column indices. Kaijser's result
is based on an application of the Furstenberg--Kesten theory to the random
matrix products $M(Y_1)M(Y_2)... M(Y_n)$. In this paper we prove a slightly
stronger form of Kaijser's theorem with a simpler argument, exploiting the
theory of e chains.
http://arXiv.org/abs/math/0702248
http://front.math.ucdavis.edu/math.PR/0702248
(alternate)
Author(s): Xun Li and Xun Yu Zhou
Abstract: This paper studies a continuous-time market where an agent, having specified
an investment horizon and a targeted terminal mean return, seeks to minimize
the variance of the return. The optimal portfolio of such a problem is called
mean-variance efficient \`{a} la Markowitz. It is shown that, when the market
coefficients are deterministic functions of time, a mean-variance efficient
portfolio realizes the (discounted) targeted return on or before the terminal
date with a probability greater than 0.8072. This number is universal
irrespective of the market parameters, the targeted return and the length of
the investment horizon.
http://arXiv.org/abs/math/0702249
http://front.math.ucdavis.edu/math.PR/0702249
(alternate)
Author(s): I. M. MacPhee and M. V. Menshikov and S. Popov and S. Volkov
Abstract: We consider an exhaustive polling system with three nodes in its transient
regime under a switching rule of generalized greedy type. We show that, for the
system with Poisson arrivals and service times with finite second moment, the
sequence of nodes visited by the server is eventually periodic almost surely.
To do this, we construct a dynamical system, the triangle process, which we
show has eventually periodic trajectories for almost all sets of parameters and
in this case we show that the stochastic trajectories follow the deterministic
ones a.s. We also show there are infinitely many sets of parameters where the
triangle process has aperiodic trajectories and in such cases trajectories of
the stochastic model are aperiodic with positive probability.
http://arXiv.org/abs/math/0702252
http://front.math.ucdavis.edu/math.PR/0702252
(alternate)
Author(s): Kurt Majewski
Abstract: We consider multiclass feedforward queueing networks with first in first out
and priority service disciplines at the nodes, and class dependent
deterministic routing between nodes. The random behavior of the network is
constructed from cumulative arrival and service time processes which are
assumed to satisfy an appropriate sample path large deviation principle. We
establish logarithmic asymptotics of large deviations for waiting time, idle
time, queue length, departure and sojourn-time processes in critical loading.
This transfers similar results from Puhalskii about single class queueing
networks with feedback to multiclass feedforward queueing networks, and
complements diffusion approximation results from Peterson. An example with
renewal inter arrival and service time processes yields the rate function of a
reflected Brownian motion. The model directly captures stationary situations.
http://arXiv.org/abs/math/0702256
http://front.math.ucdavis.edu/math.PR/0702256
(alternate)
Author(s): Thomas Simon (DP)
Abstract: Consider an inviscid Burgers equation whose initial data is a Levy a-stable
process Z with a > 1. We show that when Z has positive jumps, the Hausdorff
dimension of the set of Lagrangian regular points associated with the equation
is strictly smaller than 1/a, as soon as a is close to 1. This gives a negative
answer to a conjecture of Janicki and Woyczynski. Along the way, we contradict
a recent conjecture of Z. Shi about the lower tails of integrated stable
processes.
http://arXiv.org/abs/math/0702260
http://front.math.ucdavis.edu/math.PR/0702260
(alternate)
Author(s): David Aldous and Maxim Krikun and and Lea Popovic
Abstract: Simple stochastic models for phylogenetic trees on species have been well
studied. But much paleontology data concerns time series or trees on
higher-order taxa, and any broad picture of relationships between extant groups
requires use of higher-order taxa. A coherent model for trees on (say) genera
should involve both a species-level model and a model for the classification
scheme by which species are assigned to genera. We present a general framework
for such models, and describe three alternate classification schemes. Combining
with the species-level model of Aldous-Popovic (2005), one gets models for
higher-order trees, and we initiate analytic study of such models. In
particular we derive formulas for the lifetime of genera, for the distribution
of number of species per genus, and for the offspring structure of the tree on
genera.
http://arXiv.org/abs/q-bio/0702014
http://front.math.ucdavis.edu/q-bio.PE/0702014
(alternate)
Author(s): Martin Hairer and Etienne Pardoux
Abstract: It is well-known under the name of `periodic homogenization' that, under a
centering condition of the drift, a periodic diffusion process on R^d
converges, under diffusive rescaling, to a d-dimensional Brownian motion.
Existing proofs of this result all rely on uniform ellipticity or
hypoellipticity assumptions on the diffusion. In this paper, we considerably
weaken these assumptions in order to allow for the diffusion coefficient to
even vanish on an open set.
As a consequence, it is no longer the case that the effective diffusivity
matrix is necessarily non-degenerate. It turns out that, provided that some
very weak regularity conditions are met, the range of the effective diffusivity
matrix can be read off the shape of the support of the invariant measure for
the periodic diffusion. In particular, this gives some easily verifiable
conditions for the effective diffusivity matrix to be of full rank. We also
discuss the application of our results to the homogenization of a class of
elliptic and parabolic PDEs.
http://arXiv.org/abs/math/0702304
http://front.math.ucdavis.edu/math.PR/0702304
(alternate)
Author(s): Noam Berger and Ofer Zeitouni
Abstract: We prove that every random walk in i.i.d. environment in dimension greater
than or equal to 4 that has an almost sure positive speed in a certain
direction, an annealed invariance principle and some mild integrability
condition for regeneration times also satisfies a quenched invariance
principle. The argument is based on intersection estimates and a theorem of
Bolthausen and Sznitman.
http://arXiv.org/abs/math/0702306
http://front.math.ucdavis.edu/math.PR/0702306
(alternate)
Author(s): David Nualart (University of Kansas) and Lluis Quer-Sardanyons (Universitat Autonoma de Barcelona)
Abstract: In this paper, we extend Walsh's stochastic integral with respect to a
Gaussian noise, white in time and with some homogeneous spatial correlation, in
order to be able to integrate some random measure-valued processes. This
extension turns out to be equivalent to Dalang's one. Then we study existence
and regularity of the density of the probability law for the real-valued mild
solution to a general second order stochastic partial differential equation
driven by such a noise. For this, we apply the techniques of the Malliavin
calculus. Our results apply to the case of the stochastic heat equation in any
space dimension and the stochastic wave equation in space dimension $d=1,2,3$.
Moreover, for these particular examples, known results in the literature have
been improved.
http://arXiv.org/abs/math/0702312
http://front.math.ucdavis.edu/math.PR/0702312
(alternate)
Author(s): J. William Helton and Jean B. Lasserre and Mihai Putinar
Abstract: We investigate and discuss when the inverse of a multivariate truncated
moment matrix of a measure $\mu$ has zeros in some prescribed entries. We
describe precisely which pattern of these zeroes corresponds to independence,
namely, the measure having a product structure. A more refined finding is that
the key factor forcing a zero entry in this inverse matrix is a certain
conditional triangularity property of the orthogonal polynomials associated
with the measure $\mu$.
http://arXiv.org/abs/math/0702314
http://front.math.ucdavis.edu/math.PR/0702314
(alternate)
Author(s): Mihai Gradinaru (IECN) and Ivan Nourdin (PMA)
Abstract: The first part of the paper contains the study of the convergence for some
weighted power variations of a fractional Brownian motion B with Hurst index H
in (0,1). The behaviour is different when H<1/2 and powers are odd, compared
with the case when H=1/2 or when H>1/2 and powers are even. In the second part,
one applies the results of the first part to compute the exact rate of
convergence of some approximating schemes associated to scalar stochastic
differential equations driven by B. The limit of the error between the exact
solution and the considered scheme (whose size depends on the Hurst index H) is
computed explicitly.
http://arXiv.org/abs/math/0702317
http://front.math.ucdavis.edu/math.PR/0702317
(alternate)
Author(s): Oskar Sandberg
Abstract: Small-world graphs, which combine randomized and structured elements, are
seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this
type it is possible to route, or navigate, between vertices in few steps even
with very little knowledge of the graph itself.
We discuss a different criterion for graphs being navigable in this sense,
relating the neighbor selection of a vertex with the hitting probability of
routed walks. In several models starting from both discrete and continuous
settings, this can be showed to lead to graphs with the desired properties. It
also leads directly to a evolutionary model for the creation of similar graphs
by the stepwise rewiring of the edges, and we conjecture, supported by
simulations, that these too are navigable.
http://arXiv.org/abs/math/0702325
http://front.math.ucdavis.edu/math.PR/0702325
(alternate)
Author(s): Maria Jolis and No\`elia Viles
Abstract: We give a result of stability in law of the local time of the fractional
Brownian motion with respect to small perturbations of the Hurst parameter.
Concretely, we prove that the law (in the space of continuous functions) of the
local time of the fractional Brownian motion with Hurst parameter $H$ converges
weakly to that of the local time of $B^{H_0}$, when $H$ tends to $H_0$.
http://arXiv.org/abs/math/0702330
http://front.math.ucdavis.edu/math.PR/0702330
(alternate)
Author(s): Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
Abstract: We give sufficient conditions for tightness in the space C([0,1]) for
sequences of probability measures which enjoy a suitable decoupling between
zero level set and excursions. Applications of our results are given in the
context of (homogeneous, periodic and disordered) random walk models for
polymers and interfaces.
http://arXiv.org/abs/math/0702331
http://front.math.ucdavis.edu/math.PR/0702331
(alternate)
Author(s): E. Lytvynov and N. Ohlerich
Abstract: We construct two types of equilibrium dynamics of infinite particle systems
in a locally compact Polish space $X$, for which certain fermion point
processes are invariant. The Glauber dynamics is a birth-and-death process in
$X$, while in the case of the Kawasaki dynamics interacting particles randomly
hop over $X$. We establish conditions on generators of both dynamics under
which corresponding conservative Markov processes exist.
http://arXiv.org/abs/math/0702338
http://front.math.ucdavis.edu/math.PR/0702338
(alternate)
Author(s): {\O}yvind Ryan and M\'erouane Debbah
Abstract: Free probability and random matrix theory has shown to be a fruitful
combination in many fields of research, such as digital communications, nuclear
physics and mathematical finance. The link between free probability and
eigenvalue distributions of random matrices will be strengthened further in
this paper. It will be shown how the concept of multiplicative free convolution
can be used to express known results for eigenvalue distributions of a type of
random matrices called Information-Plus-Noise matrices. The result is proved in
a free probability framework, and some new results, useful for problems related
to free probability, are presented in this context. The connection between free
probability and estimators for covariance matrices is also made through the
notion of free deconvolution.
http://arXiv.org/abs/math/0702342
http://front.math.ucdavis.edu/math.PR/0702342
(alternate)
Author(s): Shige Peng
Abstract: The law of large numbers (LLN) and central limit theorem (CLT) are long and
widely been known as two fundamental results in probability theory.
Recently problems of model uncertainties in statistics, measures of risk and
superhedging in finance motivated us to introduce, in [4] and [5] (see also
[2], [3] and references herein), a new notion of sublinear expectation, called
\textquotedblleft% $G$-expectation\textquotedblright, and the related
\textquotedblleft$G$-normal distribution\textquotedblright from which we were
able to define G-Brownian motion as well as the corresponding stochastic
calculus. The notion of G-normal distribution plays the same important rule in
the theory of sublinear expectation as that of normal distribution in the
classic probability theory. It is then natural and interesting to ask if we
have the corresponding LLN and CLT under a sublinear expectation and, in
particular, if the corresponding limit distribution of the CLT is a G-normal
distribution. This paper gives an affirmative answer. The proof of our CLT is
short since we borrow a deep interior estimate of fully nonlinear PDE in [6]
which extended a profound result of [1] (see also [7]) to parabolic PDEs. The
assumptions of our LLN and CLT can be still improved. But the discovered
phenomenon plays the same important rule in the theory of nonlinear expectation
as that of the classical LLN and CLT in classic probability theory.
http://arXiv.org/abs/math/0702358
http://front.math.ucdavis.edu/math.PR/0702358
(alternate)
Author(s): T.R.Cass
Abstract: We consider a solution to a generic stochastic differential equation with
jumps and show that for each time the marginal law of the solution has an
infinitely differentiable density with respect to Lebesgue measure under a
uniform version of Hoermanders conditions. Our results are proved subject to
some restrictions on the rate of growth of the jump measure near zero and are
accomplished using developments of traditional arguments in Malliavin calculus.
A key ingredient in our proof is a generalisation of Norris semimartingale
inequality to discontinuous semimartingales. Unlike previous work, our results
extend beyond the case finite activity jump processes.
http://arXiv.org/abs/math/0702364
http://front.math.ucdavis.edu/math.PR/0702364
(alternate)
Author(s): Andreas Nordvall Lager{\aa}s
Abstract: Bertoin and Le Gall (2003) introduced a certain probability measure valued
Markov process that describes the evolution of a population, such that a sample
from this population would exhibit a genealogy given by the so-called
$\Lambda$-coalescent, or coalescent with multiple collisions, introduced
independently by Pitman (1999) and Sagitov (1999). We show how this process can
be extended to the case where lineages can experience mutations. Regenerative
compositions enter naturally into this model, which is somewhat surprising,
considering a negative result by M{\"o}hle (2007).
http://arXiv.org/abs/math/0702367
http://front.math.ucdavis.edu/math.PR/0702367
(alternate)
Author(s): Lev Sakhnovich
Abstract: In this article we consider the Levy processes and the corresponding
semigroup. We represent the generator of this semigroup in a convolution form.
Using the obtained convolution form and the theory of integral equations we
investigate the properties of a wide class of Levy processes (potential,
quasi-potential, the probability of the Levy process remaining within the given
domain, long time behavior, stable processes). We analyze in detail a number of
concrete examples of the Levy processes (stable processes, the variance damped
Levy processes, the variance gamma processes, the normal Gaussian process, the
Meixner process, the compound Poisson process).
http://arXiv.org/abs/math/0702378
http://front.math.ucdavis.edu/math.PR/0702378
(alternate)
Author(s): F. G\"otze and A. Tikhomirov
Abstract: We consider the joint distribution of real and imaginary parts of eigenvalues
of random matrices with independent real entries with mean zero and unit
variance. We prove the convergence of this distribution to the uniform
distribution on the unit disc without assumptions on the existence of a density
for the distribution of entries. We assume however that the entries have
sub-Gaussian tails or are sparsely non-zero.
http://arXiv.org/abs/math/0702386
http://front.math.ucdavis.edu/math.PR/0702386
(alternate)
Author(s): Francois Baccelli and Takis Konstantopoulos
Abstract: In this short paper, we consider a quadruple $(\Omega, \AA, \theta,
\mu)$,where $\AA$ is a $\sigma$-algebra of subsets of $\Omega$, and $\theta$ is
a measurable bijection from $\Omega$ into itself that preserves the measure
$\mu$. For each $B \in \AA$, we consider the measure $\mu_B$ obtained by taking
cycles (excursions) of iterates of $\theta$ from $B$. We then derive a relation
for $\mu_B$ that involves the forward and backward hitting times of $B$ by the
trajectory $(\theta^n \omega, n \in \Z)$ at a point $\omega \in \Omega$.
Although classical in appearance, its use in obtaining uniqueness of invariant
measures of various stochastic models seems to be new. We apply the concept to
countable Markov chains and Harris processes.
http://arXiv.org/abs/math/0702391
http://front.math.ucdavis.edu/math.PR/0702391
(alternate)
Author(s): J\'ozsef Balogh and B\'ela Bollob\'as and Robert Morris
Abstract: In majority bootstrap percolation on a graph G, an infection spreads
according to the following deterministic rule: if at least half of the
neighbours of a vertex v are already infected, then v is also infected, and
infected vertices remain infected forever. Percolation occurs if eventually
every vertex is infected.
The elements of the set of initially infected vertices, A \subset V(G), are
normally chosen independently at random, each with probability p, say. This
process has been extensively studied on the sequence of torus graphs [n]^d, for
n = 1,2,..., where d = d(n) is either fixed or a very slowly growing function
of n. For example, Cerf and Manzo showed that the critical probability is o(1)
if d(n) < log*(n), i.e., if p = p(n) is bounded away from zero then the
probability of percolation on [n]^d tends to one as n goes to infinity.
In this paper we study the case when the growth of d to infinity is not
excessively slow; in particular, we show that the critical probability is 1/2 +
o(1) if d > (loglog(n))^2 logloglog(n), and give much stronger bounds in the
case that G is the hypercube, [2]^d.
http://arXiv.org/abs/math/0702373
http://front.math.ucdavis.edu/math.CO/0702373
(alternate)
Author(s): Robert Berman
Abstract: We obtain various convergence results for the Bergman kernel of the Hilbert
space of all polynomials in \C^{n} of total degree at most k, equipped with a
weighted norm. The weight function is assumed to be a smooth function in \C^{n}
which grows faster than the logarithm of the squared distance function. The
convergence is studied in the large k limit and is expressed in terms of the
global equilibrium potential associated to the weight function, as well as in
terms of the Monge-Ampere measure of the weight function itself on a certain
bounded support set S. These results apply directly to the study of the
distribution of zeroes of random polynomials and of the eigenvalues of random
normal matrices.
http://arXiv.org/abs/math/0702357
http://front.math.ucdavis.edu/math.CV/0702357
(alternate)
Author(s): Jean-Pierre Dedieu and Gregorio Malajovich
Abstract: We give an upper bound in O(d ^((n+1)/2)) for the number of critical points
of a normal random polynomial. The number of minima (resp. maxima) is in
O(d^((n+1)/2)) P_n, where P_n is the (unknown) measure of the set of symmetric
positive matrices in the Gaussian Orthogonal Ensemble GOE(n). Finally, we give
a closed form expression for the number of maxima (resp. minima) of a random
univariate polynomial, in terms of hypergeometric functions.
http://arXiv.org/abs/math/0702360
http://front.math.ucdavis.edu/math.NA/0702360
(alternate)
Author(s): Amarjit Budhiraja and Arka Prasanna Ghosh
Abstract: We consider the scheduling control problem for a family of unitary networks
under heavy traffic, with general interarrival and service times, probabilistic
routing and infinite horizon discounted linear holding cost. A natural
nonanticipativity condition for admissibility of control policies is
introduced. The condition is seen to hold for a broad class of problems. Using
this formulation of admissible controls and a time-transformation technique, we
establish that the infimum of the cost for the network control problem over all
admissible sequencing control policies is asymptotically bounded below by the
value function of an associated diffusion control problem (the Brownian control
problem). This result provides a useful bound on the best achievable
performance for any admissible control policy for a wide class of networks.
http://arXiv.org/abs/math/0702402
http://front.math.ucdavis.edu/math.PR/0702402
(alternate)
Author(s): Pierre-Andr\'{e} Zitt (MODAL'X)
Abstract: In a statistical mechanics model with unbounded spins, we prove uniqueness of
the Gibbs measure under various assumptions on finite volume functional
inequalities. We follow the approach of G. Royer (1999) and obtain uniqueness
by showing convergence properties of a Glauber-Langevin dynamics. The result
was known when the measures on the box $[-n,n]^d$ (with free boundary
conditions) satisfied the same logarithmic Sobolev inequality. We generalize
this in two directions: either the constants may be allowed to grow
sub-linearly in the diameter, or we may suppose a weaker inequality than
log-Sobolev, but stronger than Poincar\'{e}. We conclude by giving a heuristic
argument showing that this could be the right inequalities to look at.
http://arXiv.org/abs/math/0702403
http://front.math.ucdavis.edu/math.PR/0702403
(alternate)
Author(s): Dirk Becherer
Abstract: We prove results on bounded solutions to backward stochastic equations driven
by random measures. Those bounded BSDE solutions are then applied to solve
different stochastic optimization problems with exponential utility in models
where the underlying filtration is noncontinuous. This includes results on
portfolio optimization under an additional liability and on dynamic utility
indifference valuation and partial hedging in incomplete financial markets
which are exposed to risk from unpredictable events. In particular, we
characterize the limiting behavior of the utility indifference hedging strategy
and of the indifference value process for vanishing risk aversion.
http://arXiv.org/abs/math/0702405
http://front.math.ucdavis.edu/math.PR/0702405
(alternate)
Author(s): W. Jaworski and C. R. E. Raja
Abstract: We obtain sufficient and necessary conditions for the Choquet-Deny theorem to
hold in the class of compactly generated totally disconnected locally compact
groups of polynomial growth, and in a larger class of totally disconnected
generalized $\ov{FC}$-groups. The following conditions turn out to be
equivalent when $G$ is a metrizable compactly generated totally disconnected
locally compact group of polynomial growth: (i) the Choquet-Deny theorem holds
for $G$; (ii) the group of inner automorphisms of $G$ acts distally on $G$;
(iii) every inner automorphism of $G$ is distal; (iv) the contraction subgroup
of every inner automorphism of $G$ is trivial; (v) $G$ is a SIN group. We also
show that for every probability measure $\mu$ on a totally disconnected
compactly generated locally compact second countable group of polynomial
growth, the Poisson boundary is a homogeneous space of $G$, and that it is a
compact homogeneous space when the support of $\mu$ generates $G$.
http://arXiv.org/abs/math/0702407
http://front.math.ucdavis.edu/math.PR/0702407
(alternate)
Author(s): Irene Klein
Abstract: The main result of the paper is a version of the fundamental theorem of asset
pricing (FTAP) for large financial markets based on an asymptotic concept of no
market free lunch for monotone concave preferences. The proof uses methods from
the theory of Orlicz spaces. Moreover, various notions of no asymptotic
arbitrage are characterized in terms of no asymptotic market free lunch; the
difference lies in the set of utilities. In particular, it is shown directly
that no asymptotic market free lunch with respect to monotone concave utilities
is equivalent to no asymptotic free lunch. In principle, the paper can be seen
as the large financial market analogue of [Math. Finance 14 (2004) 351--357]
and [Math. Finance 16 (2006) 583--588].
http://arXiv.org/abs/math/0702409
http://front.math.ucdavis.edu/math.PR/0702409
(alternate)
Author(s): Persi Diaconis and Laurent Saloff-Coste
Abstract: This paper gives a necessary and sufficient condition for a sequence of birth
and death chains to converge abruptly to stationarity, that is, to present a
cut-off. The condition involves the notions of spectral gap and mixing time. Y.
Peres has observed that for many families of Markov chains, there is a cut-off
if and only if the product of spectral gap and mixing time tends to infinity.
We establish this for arbitrary birth and death chains in continuous time when
the convergence is measured in separation and the chains all start at 0.
http://arXiv.org/abs/math/0702411
http://front.math.ucdavis.edu/math.PR/0702411
(alternate)
Author(s): Gareth O. Roberts and Jeffrey S. Rosenthal
Abstract: A $\phi$-irreducible and aperiodic Markov chain with stationary probability
distribution will converge to its stationary distribution from almost all
starting points. The property of Harris recurrence allows us to replace
``almost all'' by ``all,'' which is potentially important when running Markov
chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms
are known to be Harris recurrent. In this paper, we consider conditions under
which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are
not Harris recurrent. We present a simple but natural two-dimensional
counter-example showing how Harris recurrence can fail, and also a variety of
positive results which guarantee Harris recurrence. We also present some open
problems. We close with a discussion of the practical implications for MCMC
algorithms.
http://arXiv.org/abs/math/0702412
http://front.math.ucdavis.edu/math.PR/0702412
(alternate)
Author(s): Dmitry Kramkov and Mihai S\^{{\i}}rbu
Abstract: In the general framework of a semimartingale financial model and a utility
function $U$ defined on the positive real line, we compute the first-order
expansion of marginal utility-based prices with respect to a ``small'' number
of random endowments. We show that this linear approximation has some important
qualitative properties if and only if there is a risk-tolerance wealth process.
In particular, they hold true in the following polar cases:
\begin{tabular}@p97mm@ for any utility function $U$, if and only if the set of
state price densities has a greatest element from the point of view of
second-order stochastic dominance;for any financial model, if and only if $U$
is a power utility function ($U$ is an exponential utility function if it is
defined on the whole real line). \end{tabular}
http://arXiv.org/abs/math/0702413
http://front.math.ucdavis.edu/math.PR/0702413
(alternate)
Author(s): Andrew R. Wade
Abstract: The on-line nearest-neighbour graph on a sequence of uniform random points in
$(0,1)^d$ ($d \in \N$) joins each point after the first to its nearest
neighbour amongst its predecessors. For the total power-weighted edge length of
this graph, with weight exponent $\alpha \in (0,d/2)$, we prove a central limit
theorem (in the large-sample limit), including an expression for the limiting
variance. In contrast, we give a convergence result (with no scaling) for
$\alpha > d/2$. Both these results extend previous work. We also make some
progress in the critical case $\alpha=d/2$.
http://arXiv.org/abs/math/0702414
http://front.math.ucdavis.edu/math.PR/0702414
(alternate)
Author(s): Michael G. B. Blum and Olivier Fran\c{c}ois and Svante Janson
Abstract: For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this article,
this statistic is studied under the Yule and uniform model of phylogenetic
trees. The main tool of analysis is a coupling argument with another well-known
index called the Sackin statistic. Asymptotics for the mean, variance and
covariance of these two statistics are obtained, as well as their limiting
joint distribution for large phylogenies. Under the Yule model, the limiting
distribution arises as a solution of a functional fixed point equation. Under
the uniform model, the limiting distribution is the Airy distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and the
Catalan models for binary search trees.
http://arXiv.org/abs/math/0702415
http://front.math.ucdavis.edu/math.PR/0702415
(alternate)
Author(s): Amarjit Budhiraja and Kevin Ross
Abstract: We establish the existence of an optimal control for a general class of
singular control problems with state constraints. The proof uses weak
convergence arguments and a time rescaling technique. The existence of optimal
controls for Brownian control problems \citehar, associated with a broad family
of stochastic networks, follows as a consequence.
http://arXiv.org/abs/math/0702418
http://front.math.ucdavis.edu/math.PR/0702418
(alternate)
Author(s): Youssef Sa\"{{\i}}di and Jean-Michel Zako\"{{\i}}an
Abstract: A class of nonlinear ARCH processes is introduced and studied. The existence
of a strictly stationary and $\beta$-mixing solution is established under a
mild assumption on the density of the underlying independent process. We give
sufficient conditions for the existence of moments. The analysis relies on
Markov chain theory. The model generalizes some important features of standard
ARCH models and is amenable to further analysis.
http://arXiv.org/abs/math/0702419
http://front.math.ucdavis.edu/math.PR/0702419
(alternate)
Author(s): P. E. Ney and Anand N. Vidyashankar
Abstract: Corrections and acknowledgment for ``Local limit theory and large deviations
for supercritical branching processes'' [math.PR/0407059]
http://arXiv.org/abs/math/0702421
http://front.math.ucdavis.edu/math.PR/0702421
(alternate)
Author(s): Yuri Kifer
Abstract: Correction for Error estimates for binomial approximations of game options
[math.PR/0607123]
http://arXiv.org/abs/math/0702423
http://front.math.ucdavis.edu/math.PR/0702423
(alternate)
Author(s): Eben Kenah and James Robins
Abstract: In this paper, we outline the theory of percolation networks and their use in
the analysis of stochastic epidemic models on undirected contact networks. We
then show how the same theory can be used to analyze epidemic models with
random mixing. In the percolation network for a random-mixing model, undirected
edges disappear in the limit of a large population, so the percolation network
is purely directed. In a series of simulations, we show that percolation
networks accurately predict the mean outbreak size and probability and final
size of an epidemic for a variety of epidemic models in homogeneous and
heterogeneous populations. Finally, we show conditions under which percolation
network models are equivalent to branching processes and use percolation
networks to re-derive several classical results from different areas of
infectious disease epidemiology. In an appendix, we show how percolation
networks can be defined for any time-homogeneous stochastic epidemic model. We
conclude that the theory of percolation on semi-directed networks provides a
very general framework for the analysis of stochastic SIR epidemic models in
closed populations, which are an important part of theoretical infectious
disease epidemiology.
http://arXiv.org/abs/q-bio/0702027
http://front.math.ucdavis.edu/q-bio.QM/0702027
(alternate)
Author(s): Anne Fey and Frank Redig
Abstract: We study the rotor router model and two deterministic sandpile models. For
the rotor router model in $\mathbb{Z}^d$, Levine and Peres proved that the
limiting shape of the growth cluster is a sphere. For the other two models,
only bounds in dimension 2 are known. A unified approach for these models with
a new parameter $h$ (the initial number of particles at each site), allows to
prove a number of new limiting shape results in any dimension $d \geq 1$.
For the rotor router model, the limiting shape is a sphere for all values of
$h$. For one of the sandpile models, and $h=2d-2$ (the maximal value), the
limiting shape is a cube. For both sandpile models, the limiting shape is a
sphere in the limit $h \to -\infty$. Finally, we prove that the rotor router
shape contains a diamond, which is a new result even in the case studied by
Levine and Peres.
http://arXiv.org/abs/math/0702450
http://front.math.ucdavis.edu/math.PR/0702450
(alternate)
Author(s): Erik Ekstrom and Johan Tysk
Abstract: We study convexity and monotonicity properties for prices of bonds and bond
options when the short rate is modeled by a diffusion process. We provide
conditions under which convexity of the price in the short rate is guaranteed.
Under these conditions the price is decreasing in the drift and increasing in
the volatility of the short rate. We also study convexity properties of the
logarithm of the price.
http://arXiv.org/abs/math/0702435
http://front.math.ucdavis.edu/math.AP/0702435
(alternate)
Author(s): Huyen Pham (PMA)
Abstract: In these notes, we present some methods and applications of large deviations
to finance and insurance. We begin with the classical ruin problem related to
the Cramer's theorem and give en extension to an insurance model with
investment in stock market. We then describe how large deviation approximation
and importance sampling are used in rare event simulation for option pricing.
We finally focus on large deviations methods in risk management for the
estimation of large portfolio losses in credit risk and portfolio performance
in market investment.
http://arXiv.org/abs/math/0702473
http://front.math.ucdavis.edu/math.PR/0702473
(alternate)
Author(s): Gabor Pete
Abstract: We show that for all p>p_c(\Z^d) percolation parameters, the probability that
the cluster of the origin is finite but has at least t vertices at distance one
from the infinite cluster is exponentially small in t. Then we use this to give
a very short proof of the important fact that the isoperimetric profile of the
infinite cluster basically coincides with the profile of the original lattice.
This implies for instance that simple random walk on the largest cluster of a
finite box [-n,n]^d with high probability has L^\infty-mixing time \Theta(n^2),
and that the heat kernel (return probability) on the infinite cluster a.s.
decays like p_n(o,o)=O(n^{-d/2}). Versions of these results have been proven by
Benjamini and Mossel (2003), Mathieu and Remy (2004), Barlow (2004) and Rau
(2006). We also give a short proof of a theorem of Angel, Benjamini, Berger and
Peres (2006): the infinite percolation cluster of a wedge in \Z^3 is a.s.
transient whenever the wedge itself is transient.
http://arXiv.org/abs/math/0702474
http://front.math.ucdavis.edu/math.PR/0702474
(alternate)
Author(s): J\"{u}rgen Angst (IRMA) and Jacques Franchi (IRMA)
Abstract: Two similar Minkowskian diffusions have been considered, on one hand by
Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]),
and on the other hand by Dunkel and H\"{a}nggi ([DH1], [DH2]). We address here
the question, asked in ([DH1], [DH2]), of the asymptotic behaviour of the
variance of such diffusions. More generally, we establish a central limit
theorem for a class of Minkowskian diffusions, to which the two above ones
belong. As a consequence, we correct a partially wrong guess in [DH1].
http://arXiv.org/abs/math/0702481
http://front.math.ucdavis.edu/math.PR/0702481
(alternate)
Author(s): P. Goncalves and M.D. Jara
Abstract: For interacting particle systems that satisfies the gradient condition, the
hydrodynamic limit and the equilibrium fluctuations are well known. We prove
that under the presence of a symmetric random environment, these scaling limits
also hold for almost every choice of the environment, with homogenized
coefficients that does not depend on the particular realization of the random
environment.
http://arXiv.org/abs/math/0702513
http://front.math.ucdavis.edu/math.PR/0702513
(alternate)
Author(s): Aim\'e Lachal
Abstract: Consider the high-order heat-type equation $\partial u/\partial
t=\pm\partial^N u/\partial x^N$ for an integer $N>2$ and introduce the related
Markov pseudo-process $(X(t))_{t\ge 0}$. In this paper, we study several
functionals related to $(X(t))_{t\ge 0}$: the maximum $M(t)$ and minimum $m(t)$
up to time $t$; the hitting times $\tau_a^+$ and $\tau_a^-$ of the half lines
$(a,+\infty)$ and $(-\infty,a)$ respectively. We provide explicit expressions
for the distributions of the vectors $(X(t),M(t))$ and $(X(t),m(t))$, as well
as those of the vectors $(\tau_a^+,X(\tau_a^+))$ and $(\tau_a^-,X(\tau_a^-))$.
http://arXiv.org/abs/math/0702541
http://front.math.ucdavis.edu/math.PR/0702541
(alternate)
Author(s): Chris Howitt and Jon Warren
Abstract: The Brownian web is a random object that occurs as the scaling limit of an
infinite system of coalescing random walks. Perturbing this system of random
walks by, independently at each point in space-time, resampling the random walk
increments, leads to some natural dynamics. In this paper we consider the
corresponding dynamics for the Brownian web. In particular, pairs of coupled
Brownian webs are studied, where the second web is obtained from the first by
perturbing according to these dynamics. A stochastic flow of kernels, which we
call the erosion flow, is obtained via a filtering construction from such
coupled Brownian webs, and the N-point motions of this flow of kernels are
identified.
http://arXiv.org/abs/math/0702542
http://front.math.ucdavis.edu/math.PR/0702542
(alternate)
Author(s): K. R. Parthasarathy
Abstract: By using a quantum probabilistic approach we obtain a description of the
extreme points of the convex set of all joint probability distributions on the
product of two standard Borel spaces with fixed marginal distributions.
http://arXiv.org/abs/math/0702544
http://front.math.ucdavis.edu/math.PR/0702544
(alternate)
Author(s): Bikramjit Das and Sidney I. Resnick
Abstract: The QQ plot is a commonly used technique for informally deciding whether a
univariate random sample of size n comes from a specified distribution F. The
QQ plot graphs the sample quantiles against the theoretical quantiles of F and
then a visual check is made to see whether or not the points are close to a
straight line. For a location and scale family of distributions, the intercept
and slope of the straight line provide estimates for the shift and scale
parameters of the distribution respectively. Here we consider the set S_n of
points forming the QQ plot as a random closed set in R^2. We show that under
certain regularity conditions on the distribution F, S_n converges in
probability to a closed, non-random set. In the heavy tailed case where 1-F is
a regularly varying function, a similar result can be shown but a modification
is necessary to provide a statistically sensible result since typically F is
not completely known.
http://arXiv.org/abs/math/0702551
http://front.math.ucdavis.edu/math.PR/0702551
(alternate)
Author(s): Tomasz Schreiber and Joseph E. Yukich
Abstract: We show that the random point measures induced by vertices in the convex hull
of a Poisson sample on the unit ball, when properly scaled and centered,
converge to those of a mean zero Gaussian field. We establish limiting variance
and covariance asymptotics in terms of the density of the Poisson sample.
Similar results hold for the point measures induced by the maximal points in a
Poisson sample. The approach involves introducing a generalized spatial birth
growth process allowing for cell overlap.
http://arXiv.org/abs/math/0702553
http://front.math.ucdavis.edu/math.PR/0702553
(alternate)
Author(s): Guilhem Coq (1) and Olivier Alata (2) and Marc Arnaudon (1) and Christian Olivier (2) ((1) Laboratoire de Math\'ematiques et Applications Poitiers
France, (2) Laboratoire Signal Image et Communications Poitiers France)
Abstract: Information criteria are an appropriate and widely used tool for solving
model selection problems. However, different ways to use them exist, each
leading to a more or less precise approximation of the sought model. In this
paper, we mainly present two methods of utilisation of information criteria :
the classical one which is generally used and an alternative one, more precise
but requiring a little more calculations. Those methods are compared on 1-D and
2-D autoregressive models ; we use a synthetized process for the 1-D case and
texture images for the 2-D case. We also work with the original phi_beta
criterion which includes all others usual criteria such as AIC, BIC, and phi.
http://arXiv.org/abs/math/0702540
http://front.math.ucdavis.edu/math.ST/0702540
(alternate)
Author(s): Gautam Iyer
Abstract: We consider the incompressible Navier-Stokes equations with spatially
periodic boundary conditions. If the Reynolds number is small enough we provide
an elementary short proof of the existence of global in time H\"older
continuous solutions. Our proof is based on the stochastic Lagrangian
formulation of the Navier-Stokes equations, and works in both the two and three
dimensional situation.
http://arXiv.org/abs/math/0702506
http://front.math.ucdavis.edu/math.AP/0702506
(alternate)
Author(s): L. Caramellino and B. Pacchiarotti
Abstract: The paper deals with the asymptotic behavior of the bridge of a Gaussian
process conditioned to stay in $n$ fixed points at $n$ fixed past instants. In
particular, functional large deviation results are stated for small time.
Several examples are considered: integrated or not fractional Brownian motion,
$m$-fold integrated Brownian motion. As an application, the asymptotic behavior
of the exit probability is studied and used for the practical purpose of the
numerical computation, via Monte Carlo methods, of the hitting probability up
to a given time.
http://arXiv.org/abs/math/0702573
http://front.math.ucdavis.edu/math.PR/0702573
(alternate)
Author(s): Carlo Marinelli
Abstract: We give sufficient conditions for existence, uniqueness and ergodicity of
invariant measures for Musiela's stochastic partial differential equation with
deterministic volatility and a Hilbert space valued driving Levy noise.
Conditions for the absence of arbitrage and for the existence of mild solutions
are also discussed.
http://arXiv.org/abs/math/0702622
http://front.math.ucdavis.edu/math.PR/0702622
(alternate)
Author(s): S. Gerhold
Abstract: We show that the probability mass function of the riff-shuffle distribution,
also known as the minimum negative binomial distribution, is unimodal, but in
general not log-concave.
http://arXiv.org/abs/math/0702639
http://front.math.ucdavis.edu/math.PR/0702639
(alternate)
Author(s): Julien Barral and Benoit Mandelbrot
Abstract: The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq
2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the
unit interval into $b$ subintervals and multiplies the density in each
subinterval by either 1 or -1 with the respective frequencies of $\frac{1%
}{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the resulting
density can be renormalized so that, as $n\to \infty $ ($n$ being the number of
iterations) the signed measure converges in some sense to a non-degenerate
limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$, renormalization
creates a martingale, the convergence is strong, and the limit shares the
H\"{o}lder and Hausdorff properties of the fractional Brownian motion of
exponent $H$. If $H\leq {1}/{2}$, hence $p\leq b^{{-1}/{2}%}$, this martingale
does not converge. However, a different normalization can be applied, for
$H\leq {1/2}$ to the martingale itself and for $H>% {1/2}$ to the discrepancy
between the limit and a finite approximation. In all cases the resulting
process is found to converge weakly to the Wiener Brownian motion,
independently of $H$ and of $b$. Thus, to the usual additive paths toward
Wiener measure, this procedure adds an infinity of multiplicative paths.
http://arXiv.org/abs/math/0702644
http://front.math.ucdavis.edu/math.PR/0702644
(alternate)
Author(s): Doug Pickrell
Abstract: The universal covering of the group PSU(1,1) acts naturally on H^m(\delta),
the space of holomorphic differentials of order m on the Poincare disk. The
purpose of this paper is to survey, as broadly as I am able, the basic sources
and examples of invariant measures for this action.
http://arXiv.org/abs/math/0702672
http://front.math.ucdavis.edu/math.PR/0702672
(alternate)
Author(s): Akihiko Inoue
Abstract: We prove a representation of the partial autocorrelation function (PACF), or
the Verblunsky coefficients, of a stationary process in terms of the AR and MA
coefficients. We apply it to show the asymptotic behaviour of the PACF. We also
propose a new definition of short and long memory in terms of the PACF.
http://arXiv.org/abs/math/0702648
http://front.math.ucdavis.edu/math.SP/0702648
(alternate)
Author(s): M. Gregoratti
Abstract: We study the Classical Probability analogue of the dilations of a quantum
dynamical semigroup in Quantum Probability. Given a (not necessarily
homogeneous) Markov chain in discrete time in a finite state space E, we
introduce a second system, an environment, and a deterministic invertible
time-homogeneous global evolution of the system E with this environment such
that the original Markov evolution of E can be realized by a proper choice of
the initial random state of the environment. We also compare this dilations
with the dilations of a quantum dynamical semigroup in Quantum Probability:
given a classical Markov semigroup, we show that it can be extended to a
quantum dynamical semigroup for which we can find a quantum dilation to a group
of *-automorphisms admitting an invariant abelian subalgebra where this quantum
dilation gives just our classical dilation.
http://arXiv.org/abs/math/0702690
http://front.math.ucdavis.edu/math.PR/0702690
(alternate)
Author(s): Tomasz Bojdecki and Luis G. Gorostiza and Anna Talarczyk
Abstract: In this paper we study three self-similar, long-range dependence, Gaussian
processes. The first one, with covariance
\int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1
http://arXiv.org/abs/math/0702708
http://front.math.ucdavis.edu/math.PR/0702708
(alternate)
Author(s): Robert C. Dalang and Davar Khoshnevisan and and Eulalia Nualart
Abstract: We consider a system of $d$ coupled non-linear stochastic heat equations in
spatial dimension 1 driven by $d$-dimensional additive space-time white noise.
We establish upper and lower bounds on hitting probabilities of the solution
$\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of respectively
Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff
dimensions of level sets and their projections. A result of independent
interest is an anisotropic form of the Kolmogorov continuity theorem.
http://arXiv.org/abs/math/0702710
http://front.math.ucdavis.edu/math.PR/0702710
(alternate)
Author(s): J. Dony and D. M. Mason
Abstract: In 1991 Stute introduced a class of estimators called conditional
U-statistics. They can be seen as a generalization of the Nadaraya-Watson
estimator, and their strong pointwise consistency to the general regression
function has been obtained in the same paper by Stute. Very recently, Gine and
Mason introduced the notion of a local U-process, which generalizes that of a
local empirical process, and obtained central limit theorems and laws of the
iterated logarithm for this class. We apply the methods developed by Einmahl
and Mason (2005) and Gine and Mason (2007a,b) to establish uniform in bandwidth
consistency to the general regression function of the estimator proposed by
Stute.
http://arXiv.org/abs/math/0702696
http://front.math.ucdavis.edu/math.ST/0702696
(alternate)
Author(s): Zeev Rudnick and Igor Wigman
Abstract: We study the volume of nodal sets for eigenfunctions of the Laplacian on the
standard torus in two or more dimensions. We consider a sequence of eigenvalues
$4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the
expectation and variance of the volume of the nodal set with respect to a
Gaussian probability measure on the eigenspaces. We show that the expected
volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main result is that
the variance of the volume normalized by $\sqrt{\eigenvalue}$ is bounded by
$O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing fluctuations
as we increase the dimension of the eigenspace.
http://arXiv.org/abs/math-ph/0702081
http://front.math.ucdavis.edu/math-ph/0702081
(alternate)
Author(s): Traian A Pirvu and Ulrich G Haussmann
Abstract: This paper derives a portfolio decomposition formula when the agent maximizes
utility of her wealth at some finite planning horizon. The financial market is
complete and consists of multiple risky assets (stocks) plus a risk free asset.
The stocks are modelled as exponential Brownian motions with drift and
volatility being Ito processes. The optimal portfolio has two components: a
myopic component and a hedging one. We show that the myopic component is robust
with respect to stopping times. We employ the Clark-Haussmann formula to derive
portfolio s hedging component.
http://arXiv.org/abs/math/0702726
http://front.math.ucdavis.edu/math.PR/0702726
(alternate)
Author(s): Traian A Pirvu and Ulrich G Haussmann
Abstract: This paper studies the problem of optimal investment in incomplete markets,
robust with respect to stopping times. We work on a Brownian motion framework
and the stopping times are adapted to the Brownian filtration. Robustness can
only be achieved for logartihmic utility, otherwise a cashflow should be added
to the investor s wealth. The cashflow can be decomposed into the sum of an
increasing and a decreasing process. The last one can be viewed as consumption.
The first one is an insurance premium the agent has to pay.
http://arXiv.org/abs/math/0702727
http://front.math.ucdavis.edu/math.PR/0702727
(alternate)
Author(s): Martin Dyer and Leslie Ann Goldberg and Mark Jerrum
Abstract: We give a systematic development of the application of matrix norms to rapid
mixing in spin systems. We show that rapid mixing of both random update Glauber
dynamics and systematic scan Glauber dynamics occurs if any matrix norm of the
associated dependency matrix is less than 1. We give improved analysis for the
case in which the diagonal of the dependency matrix is 0 (as in heat bath
dynamics). We apply the matrix norm methods to random update and systematic
scan Glauber dynamics for colouring various classes of graphs. We give a
general method for estimating a norm of a symmetric non-regular matrix. This
leads to improved mixing times for any class of graphs which is hereditary and
sufficiently sparse including several classes of degree-bounded graphs such as
non-regular graphs, trees, planar graphs and graphs with given tree-width and
genus.
http://arXiv.org/abs/math/0702744
http://front.math.ucdavis.edu/math.PR/0702744
(alternate)
Author(s): Geoffrey Pritchard and Mark C. Wilson
Abstract: We consider the problem of manipulation of elections using positional voting
rules under Impartial Culture voter behaviour. We consider both the logical
possibility of coalitional manipulation, and the number of voters that must be
recruited to form a manipulating coalition. It is shown that the manipulation
problem may be well approximated by a very simple linear program in two
variables. This permits a comparative analysis of the asymptotic
(large-population) manipulability of the various rules. It is seen that the
manipulation resistance of positional rules with 5 or 6 (or more) candidates is
quite different from the more commonly analyzed 3- and 4-candidate cases.
http://arXiv.org/abs/math/0702752
http://front.math.ucdavis.edu/math.PR/0702752
(alternate)
Author(s): M. Gregoratti
Abstract: We study the Classical Probability analogue of the dilations of a quantum
dynamical semigroup defined in Quantum Probability via quantum stochastic
differential equations. Given a homogeneous Markov chain in continuous time in
a finite state space E, we introduce a second system, an environment, and a
deterministic invertible time-homogeneous global evolution of the system E with
this environment such that the original Markov evolution of E can be realized
by a proper choice of the initial random state of the environment. We also
compare this dilations with the dilations of a quantum dynamical semigroup in
Quantum Probability: given a classical Markov semigroup, we extend it to a
proper quantum dynamical semigroup for which we can find a Hudson-Parthasarathy
dilation which is itself an extension of our classical dilation.
http://arXiv.org/abs/math/0702784
http://front.math.ucdavis.edu/math.PR/0702784
(alternate)
Author(s): Larbi Alili and Ching-Tang Wu
Abstract: A class of Volterra transforms, preserving the Wiener measure, with kernels
of Goursat type is considered. We provide some results on the inverses of the
associated Gramian matrices. These are applied to the study of a class of
linear singular stochastic differential equations together with the
corresponding decompositions of filtrations. The studied equations are viewed
as non-canonical decompositions of some generalized bridges.
http://arXiv.org/abs/math/0702785
http://front.math.ucdavis.edu/math.PR/0702785
(alternate)
Author(s): Joan-Andreu L\'azaro-Cam\'{\i} and Juan-Pablo Ortega
Abstract: We use the global stochastic analysis tools introduced by P. A. Meyer and L.
Schwartz to write down a stochastic generalization of the Hamilton equations on
a Poisson manifold that, for exact symplectic manifolds, satisfy a natural
critical action principle similar to the one encountered in classical
mechanics. Several features and examples in relation with the solution
semimartingales of these equations are presented.
http://arXiv.org/abs/math/0702787
http://front.math.ucdavis.edu/math.PR/0702787
(alternate)
Author(s): Yueyun Hu (LAGA) and Zhan Shi (PMA)
Abstract: We establish a second-order almost sure limit theorem for the minimal
position in a one-dimensional super-critical branching random walk, and also
prove a martingale convergence theorem which answers a question of Biggins and
Kyprianou (2005). Our method applies furthermore to the study of directed
polymers on a disordered tree; in particular, we give a rigorous proof of a
phase transition phenomenon for the partition function, described by Derrida
and Spohn (1988).
http://arXiv.org/abs/math/0702799
http://front.math.ucdavis.edu/math.PR/0702799
(alternate)
Author(s): Fumio Hiai and Takuho Miyamoto and Yoshimichi Ueda
Abstract: Motivated by Voiculescu's liberation theory, we introduce the orbital free
entropy $\chi_orb$ for non-commutative self-adjoint random variables (also for
"hyperfinite random multivariables"). Besides its basic properties the relation
of $\chi_orb$ with the usual free entropy $\chi$ is shown. Moreover, the
dimension counterpart of $\chi_orb$ is discussed.
http://arXiv.org/abs/math/0702745
http://front.math.ucdavis.edu/math.OA/0702745
(alternate)
Author(s): Louis H. Y. Chen and Aihua Xia
Abstract: This exposition explains the basic ideas of Stein's method for Poisson random
variable approximation and Poisson process approximation from the point of view
of the immigration-death process and Palm theory. The latter approach also
enables us to define local dependence of point processes [Chen and Xia (2004)]
and use it to study Poisson process approximation for locally dependent point
processes and for dependent superposition of point processes.
http://arXiv.org/abs/math/0702820
http://front.math.ucdavis.edu/math.PR/0702820
(alternate)
Author(s): Tzuu-Shuh Chiang and Shang-Yuan Shiu and Shuenn-Jyi Sheu
Abstract: In a market with transaction costs, the price of a derivative can be
expressed in terms of (preconsistent) price systems (after Kusuoka (1995)). In
this paper, we consider a market with binomial model for stock price and
discuss how to generate the price systems. From this, the price formula of a
derivative can be reformulated as a stochastic control problem. Then the
dynamic programming approach can be used to calculate the price. We also
discuss optimization of expected utility using price systems.
http://arXiv.org/abs/math/0702828
http://front.math.ucdavis.edu/math.PR/0702828
(alternate)
Author(s): Dmitry B. Rokhlin
Abstract: This paper deals with the notion of a large financial market and the concepts
of asymptotic arbitrage and strong asymptotic arbitrage (both of the first
kind), introduced by Yu.M. Kabanov and D.O. Kramkov. We show that the arbitrage
properties of a large market are completely determined by the asymptotic
behavior of the sequence of the num\'eraire portfolios, related to the small
markets. The obtained criteria can be expressed in terms of contiguity, entire
separation and Hellinger integrals, provided these notions are extended to
sub-probability measures. As examples we consider market models on finite
probability spaces, semimartingale and diffusion models. Also a discrete-time
infinite horizon market model with one log-normal stock is examined.
http://arXiv.org/abs/math/0702849
http://front.math.ucdavis.edu/math.PR/0702849
(alternate)
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