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Probability Abstracts 88
This document contains abstracts 3516-3646.
They have been mailed on September 1, 2005.
Author(s): Akihiko Inoue and Yumiharu Nakano
Abstract: We consider an investment model with memory in which the prices of n risky
assets are driven by an n-dimensional Gaussian process with stationary
increments that is different from Brownian motion. The driving process consists
of n independent components, and each component is characterized by two
parameters describing the memory. For the model, we explicitly solve the
problem of maximizing the expected growth rate as well as that of maximizing
the probability of overperforming a given benchmark.
http://arXiv.org/abs/math/0506621
http://front.math.ucdavis.edu/math.PR/0506621
(alternate)
Author(s): Vladimir I. Bogachev and Giuseppe Da Prato and Michael R\"ockner and Zeev Sobol
Abstract: Let $L$ be a second order elliptic operator on $R^d$ with a constant
diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$
with some $p>d$.
We assume that $L$ possesses a Lyapunov function, but no local boundedness of
$b$ is assumed. It is known that then there exists a unique probability measure
$\mu$ satisfying the equation $L^*\mu=0$ and that the closure of $L$ in
$L^1(\mu)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the resolvent
$\{G_\lambda\}_{\lambda > 0}$.
We prove that, for any Lipschitzian function $f\in L^1(\mu)$ and all
$t,\lambda>0$, the functions $T_tf$ and $G_\lambda f$ are Lipschitzian and
|\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_\lambda f(x)| \leq
\frac{1}{\lambda} G_\lambda |\nabla f|(x).
An analogous result is proved in the parabolic case.
http://arXiv.org/abs/math/0507079
http://front.math.ucdavis.edu/math.PR/0507079
(alternate)
Author(s): Alan Hammond and Fraydoun Rezakhanlou
Abstract: We study a model of mass-bearing coagulating planar Brownian particles.
Coagulation is prone to occur when two particles become within a distance of
order $\epsilon$. We assume that the initial number of particles is of the
order of $| \log \epsilon |. Under suitable assumptions on the initial
distribution of particles and the microscopic coagulation propensities, we show
that the macroscopic particle densities satisfy a Smoluchowski-type equation.
http://arXiv.org/abs/math/0507522
http://front.math.ucdavis.edu/math.PR/0507522
(alternate)
Author(s): Jason Swanson
Abstract: We consider the median of n independent Brownian motions, and show that this
process, when properly scaled, converges weakly to a centered Gaussian process.
The chief difficulty is establishing tightness, which is proved through direct
estimates on the increments of the median process. An explicit formula is given
for the covariance function of the limit process. The limit process is also
shown to be Holder continuous with exponent gamma for all gamma < 1/4.
http://arXiv.org/abs/math/0507524
http://front.math.ucdavis.edu/math.PR/0507524
(alternate)
Author(s): Sourav Chatterjee
Abstract: The purpose of this dissertation is to introduce a version of Stein's method
of exchangeable pairs to solve problems in measure concentration. We
specifically target systems of dependent random variables, since that is where
the power of Stein's method is fully realized. Because the theory is quite
abstract, we have tried to put in as many examples as possible. Some of the
highlighted applications are as follows: (a) We shall find an easily verifiable
condition under which a popular heuristic technique originating from physics,
known as the ``mean field equations'' method, is valid. No such condition is
currently known. (b) We shall present a way of using couplings to derive
concentration inequalities. Although couplings are routinely used for proving
decay of correlations, no method for using couplings to derive concentration
bounds is available in the literature. This will be used to obtain (c)
concentration inequalities with explicit constants under Dobrushin's condition
of weak dependence. (d) We shall give a method for obtaining concentration of
Haar measures using convergence rates of related random walks on groups. Using
this technique and one of the numerous available results about rates of
convergence of random walks, we will then prove (e) a quantitative version of
Voiculescu's celebrated connection between random matrix theory and free
probability.
http://arXiv.org/abs/math/0507526
http://front.math.ucdavis.edu/math.PR/0507526
(alternate)
Author(s): S Satheesh and E Sandhya and S Sherly
Abstract: Here we develop a first order autoregressive model {Xn} that is marginally
stationary where Xn is the sum/ extreme of k i.i.d observations. We prove that
stationary solutions to these models are either semi- selfdecomposable/
extreme-semi-selfdecomposable or, sum/ extreme stable with respect to Harris
distribution.
http://arXiv.org/abs/math/0507535
http://front.math.ucdavis.edu/math.PR/0507535
(alternate)
Author(s): Leonid Mytnik and Edwin Perkins and Anja Sturm
Abstract: We consider the existence and pathwise uniqueness of the stochastic heat
equation with a multiplicative colored noise term on IR^d for d greater or
equal to 1. We focus on the case of non-Lipschitz noise coefficients and
singular spatial noise correlations. In the course of the proof a new result on
Hoelder continuity of the solutions near zero is established.
http://arXiv.org/abs/math/0507545
http://front.math.ucdavis.edu/math.PR/0507545
(alternate)
Author(s): Gregory J. Morrow and Yu Zhang
Abstract: Let L_n denote the lowest crossing of the 2n \times 2n square box B(n)
centered at the origin for critical site percolation on Z^2 or critical site
percolation on the triangular lattice imbedded in Z^2, and denote by Q_n the
set of pivotal sites along this crossing. On the event that a pivotal site
exists, denote the minimum height that a pivotal site attains above the bottom
of B(n) by M_n:= min{m:(x,-n+m)\in Q_n for some -n\le x\le n}. Else, define M_n
= 2n. We prove that P(M_n < m) \asymp m/n, uniformly for 1\le m\le n. This
relation extends Theorem 1 of van den Berg and Jarai (2003) who handle the
corresponding distribution for the lowest crossing in a slightly different
context. As a corollary we establish the asymptotic distribution of the minimum
height of the set of cut points of a certain chordal SLE_6 in the unit square
of C.
http://arXiv.org/abs/math/0507566
http://front.math.ucdavis.edu/math.PR/0507566
(alternate)
Author(s): Adam Paszkiewicz
Abstract: We characterize sequences of numbers $(a_n)$ such that $\sum_{n\geq 1}
a_n\Phi_n$ converges a.e. for any orthonormal system $(\Phi_n)$ in any
$L_2$-space. In our criterion, we use the set $B =\{\sum_{m\geq n} |a_m|^2;
n\geq 1\}$ and its information function $$h_B(t) = -\log_3(\beta-\alpha)$$ for
$t\in (\alpha, \beta]$, $[\alpha, \beta]\cap B =\{\alpha, \beta\}.$
http://arXiv.org/abs/math/0507568
http://front.math.ucdavis.edu/math.AP/0507568
(alternate)
Author(s): R. Douc (\'Ecole Polytechnique and Palaiseau) and France E. Moulines (\'Ecole Nationale Sup\'erieure des T\'el\'ecommunications, Paris)
Abstract: In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key
tool in computational statistics. These algorithms approximate a sequence of
distributions by a sequence of weighted empirical measures associated to a
weighted population of particles. These particles and weights are generated
recursively according to elementary transformations: mutation and selection.
Examples of applications include the sequential Monte-Carlo techniques to solve
optimal non-linear filtering problems in state-space models, molecular
simulation, genetic optimization, etc.
Despite many theoretical advances the asymptotic property of these
approximations remains of course a question of central interest. In this paper,
we analyze sequential Monte Carlo methods from an asymptotic perspective, that
is, we establish law of large numbers and invariance principle as the number of
particles gets large. We introduce the concepts of "weighted sample"
consistency and asymptotic normality, and derive conditions under which the
mutation and the selection procedure used in the sequential Monte-Carlo
build-up preserve these properties. To illustrate our findings, we analyze SMC
algorithms to approximate the filtering distribution in state-space models. We
show how our techniques allow to relax restrictive technical conditions used in
previously reported works and provide grounds to analyze more sophisticated
sequential sampling strategies.
http://arXiv.org/abs/math/0507042
http://front.math.ucdavis.edu/math.ST/0507042
(alternate)
Author(s): J.M. Swart
Abstract: This paper considers contact processes on general lattices. Assuming that the
expected number of infected sites grows subexponentially, it is shown that the
configuration as seen from a typical (`Palmed') infected site at an
exponentially distributed time converges, as time tends to infinity, to the
upper invariant law conditioned on the origin being infected. The assumption
that the expected number of infected sites grows subexponentially is shown to
be satisfied if the lattice has subexponential growth and the infection rates
satisfy an exponential moment condition.
http://arXiv.org/abs/math/0507578
http://front.math.ucdavis.edu/math.PR/0507578
(alternate)
Author(s): Krzysztof Bogdan and Pawe{\l} Sztonyk
Abstract: We characterize those homogeneous translation invariant symmetric non-local
operators with positive maximum principle whose harmonic functions satisfy
Harnack's inequality. We also estimate the corresponding semigroup and the
potential kernel.
http://arXiv.org/abs/math/0507579
http://front.math.ucdavis.edu/math.PR/0507579
(alternate)
Author(s): Sebastien Blachere and Sara Brofferio
Abstract: The Internal Diffusion Limited Aggregation has been introduced by Diaconis
and Fulton in 1991. It is a growth model defined on an infinite set and
associated to a Markov chain on this set. We focus here on sets which are
finitely generated groups with exponential growth. We prove a shape theorem for
the Internal DLA on such groups associated to symmetric random walks. For that
purpose, we introduce a new distance associated to the Green function, which
happens to have some interesting properties. In the case of homogeneous trees,
we also get the right order for the fluctuations of that model around its
limiting shape.
http://arXiv.org/abs/math/0507582
http://front.math.ucdavis.edu/math.PR/0507582
(alternate)
Author(s): J. Gaertner and W. Koenig and S. Molchanov
Abstract: We consider the parabolic Anderson problem $\partial_t u =\Delta u+\xi(x) u$
on $\R_+\times \Z^d$ with localized initial condition $u(0,x)=\delta_0(x)$ and
random i.i.d. potential $\xi$. Under the assumption that the distribution of
$\xi(0)$ lies in the vicinity of, or beyond, the double-exponential
distribution, we prove the following geometric characterisation of
intermittency: with probability one, as $t\to\infty$, the overwhelming
contribution to the total mass $\sum_x u(t,x)$ comes from a slowly increasing
number of islands which are located far from each other. These islands are
local regions of those high exceedances of the field $\xi$ in a box with radius
$t\log^2t$ for which the (local) principal Dirichlet eigenvalue of the random
operator $\Delta+\xi$ is close to maximal. We also prove that the shape of
$\xi$ in these regions is non-random and that $u(t,\cdot)$ is close to the
corresponding positive eigenfunction. This is the geometric picture suggested
by localization theory for the Anderson Hamiltonian.
http://arXiv.org/abs/math/0507585
http://front.math.ucdavis.edu/math.PR/0507585
(alternate)
Author(s): Rui Dong and Christina Goldschmidt and James B. Martin
Abstract: In this paper we give a new example of duality between fragmentation and
coagulation operators. Consider the space of partitions of mass (that is,
decreasing sequences of non-negative real numbers whose sum is 1) and the
two-parameter family of Poisson-Dirichlet distributions PD(alpha,theta), taking
values in this space. We introduce families of random fragmentation and
coagulation operators, Frag_{alpha} and Coag_{alpha,theta} respectively, with
the following property: if the input to Frag_{alpha} has PD(alpha,theta)
distribution then the output has PD(alpha,theta+1) distribution, while the
reverse is true for Coag_{alpha,theta}. This result may be proved using a
subordinator representation, and provides a companion set of relations to those
of Pitman between PD(alpha,theta) and PD(alpha*beta,theta). Repeated
application of the Frag_{alpha} operators gives rise to a family of
fragmentation chains. We show that these Markov chains can be encoded
natuarally by certain random recursive trees, and use this representation to
give an alternative and more concrete proof of the coagulation-fragmentation
duality.
http://arXiv.org/abs/math/0507591
http://front.math.ucdavis.edu/math.PR/0507591
(alternate)
Author(s): Jens Marklof and Yves Tourigny and Lech Wolowski
Abstract: We construct explicit invariant measures for a family of infinite products of
random, independent, identically-distributed elements of SL(2,C). The matrices
in the product are such that one entry is gamma-distributed along a ray in the
complex plane. When the ray is the positive real axis, the products are those
associated with a continued fraction studied by Letac and Seshadri [Z. Wahr.
Verw. Geb. 62 (1983) 485-489], who showed that the distribution of the
continued fraction is a generalised inverse Gaussian. We extend this result by
finding the distribution for an arbitrary ray in the complex right-half plane,
and thus compute the corresponding Lyapunov exponent explicitly. When the ray
lies on the imaginary axis, the matrices in the infinite product coincide with
the transfer matrices associated with a one-dimensional discrete Schroedinger
operator with a random, gamma-distributed potential. Hence, the explicit
knowledge of the Lyapunov exponent may be used to estimate the (exponential)
rate of localisation of the eigenstates.
http://arXiv.org/abs/math-ph/0507069
http://front.math.ucdavis.edu/math-ph/0507069
(alternate)
Author(s): Gleb Yakovlev and John B. Rundle and Robert Shcherbakov and and Donald L. Turcotte
Abstract: We derive an analytical expression of the inter-arrival time distribution for
a non-homogeneous Poisson process (NHPP). This expression is exact and is
applicable to any time interval, finite or infinite. As an illustration, we
present simulation results for three different intensity functions.
http://arXiv.org/abs/cond-mat/0507657
http://front.math.ucdavis.edu/cond-mat/0507657
(alternate)
Author(s): Tom Kennedy
Abstract: The Schramm-Loewner evolution (SLE) can be simulated by dividing the time
interval into N subintervals and approximating the random conformal map of the
SLE by the composition of N random, but relatively simple, conformal maps. In
the usual implementation the time required to compute a single point on the SLE
curve is O(N). We give an algorithm for which the time to compute a single
point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a
value of p of approximately 0.4.
http://arXiv.org/abs/math/0508002
http://front.math.ucdavis.edu/math.PR/0508002
(alternate)
Author(s): Karen Ball and Tom Kurtz and Lea Popovic and and Greg Rempala
Abstract: A reaction network is a chemical system involving multiple reactions and
chemical species. Stochastic models of such networks treat the system as a
continuous time Markov chain on the number of molecules of each species with
reactions as possible transitions of the chain. In many cases of biological
interest some of the chemical species in the network are present in much
greater abundance than others and reaction rate constants can vary over several
orders of magnitude. We consider approaches to approximation of such models
that take the multiscale nature of the system into account. Our primary example
is a model of a cell's viral infection for which we apply a combination of
averaging and law of large number arguments to show that the ``slow'' component
of the model can be approximated by a deterministic equation and to
characterize the asymptotic distribution of the ``fast'' components. The main
goal is to illustrate techniques that can be used to reduce the dimensionality
of much more complex models.
http://arXiv.org/abs/math/0508015
http://front.math.ucdavis.edu/math.PR/0508015
(alternate)
Author(s): Ivan Werner
Abstract: We continue the study of Markov systems started in \cite{Wer1}. In this
paper, we prove a generalization of Breiman's strong low of large numbers
\cite{Br} which implies a necessary condition for the uniqueness of the
stationary state of a Markov system.
http://arXiv.org/abs/math/0508054
http://front.math.ucdavis.edu/math.PR/0508054
(alternate)
Author(s): Luc Bouten and Ramon van Handel
Abstract: These notes are intended as an introduction to noncommutative (quantum)
filtering theory. An introduction to quantum probability theory is given,
focusing on the spectral theorem and the conditional expectation as the least
squares estimate, and culminating in the construction of Wiener and Poisson
processes on the Fock space. Next we describe the Hudson-Parthasarathy quantum
Ito calculus and its use in the modelling of physical systems. Finally, we use
a reference probability method to obtain quantum filtering equations, in the
Belavkin-Zakai (unnormalized) form, for several system-observation models from
quantum optics. The normalized (Belavkin-Kushner-Stratonovich) form is obtained
through a noncommutative analogue of the Kallianpur-Striebel formula.
http://arXiv.org/abs/math-ph/0508006
http://front.math.ucdavis.edu/math-ph/0508006
(alternate)
Author(s): Mathieu Bouville
Abstract: Using certain techniques a billiards player can have long series of easy
shots --each shot leading to another easy shot-- and very high scores. As the
usual model for carom billiards assumes a Bernoulli process which does not
account for such correlations, it cannot capture this important feature of the
game. Modelling carom billiards as a Markov process, the probability to make a
shot can be made to depend on the previous shot. The improved agreement with
data is an indication that a Markov process indeed captures the effects of
position play better. Moreover it is possible to quantify how much a player
plays position. Given two players with the same average, one can tell the good
shot-maker from the good position player. This can be useful for players (and
their coaches) to evaluate their strengths and weaknesses.
http://arXiv.org/abs/math/0508089
http://front.math.ucdavis.edu/math.PR/0508089
(alternate)
Author(s): Christopher Hoffman
Abstract: We consider a wide class of ergodic first passage percolation processes on
Z^2 and prove that there exist at least four one-sided geodesics a.s. We also
show that coexistence is possible with positive probability in a four color
Richardson's growth model. This improves earlier results of Haggstrom and
Pemantle, Garet and Marchand, and Hoffman who proved that first passage
percolation has at least two geodesics and that coexistence is possible in a
two color Richardson's growth model.
http://arXiv.org/abs/math/0508114
http://front.math.ucdavis.edu/math.PR/0508114
(alternate)
Author(s): Richard Kenyon and Andrei Okounkov
Abstract: In this paper we study surfaces in R^3 that arise as limit shapes in a class
of random surface models arising from dimer models. The limit shapes are
minimizers of a surface tension functional, that is, they minimize, for fixed
boundary conditions, the integral of a quantity (the surface tension) depending
only on the slope of the surface. The surface tension as a function of the
slope has singularities and is not strictly convex, which leads to formation of
facets and edges in the limit shapes.
We find a change of variables that reduces the Euler-Lagrange equation for
the variational problem to the complex inviscid Burgers equation (complex Hopf
equation). The equation can thus be solved in terms of an arbitrary holomorphic
function, which is somewhat similar in spirit to Weierstrass parametrization of
minimal surfaces. We further show that for a natural dense set of boundary
conditions, the holomorphic function in question is, in fact, algebraic. The
tools of algebraic geometry can thus be brought in to study the the minimizers
and, especially, the formation of their singularities. This is illustrated by
several explicitly computed examples.
http://arXiv.org/abs/math-ph/0507007
http://front.math.ucdavis.edu/math-ph/0507007
(alternate)
Author(s): Mark Conger
Abstract: Given a permutation $\pi$ chosen uniformly from $S_n$, we explore the joint
distribution of $\pi(1)$ and the number of descents in $\pi$. We obtain a
formula for the number of permutations with $\des(\pi)=d$ and $\pi(1)=k$, and
use it to show that if $\des(\pi)$ is fixed at $d$, then the expected value of
$\pi(1)$ is $d+1$. We go on to derive generating functions for the joint
distribution, show that it is unimodal if viewed correctly, and show that when
$d$ is small the distribution of $\pi(1)$ among the permutations with $d$
descents is approximately geometric. Applications to Stein's method and the
Neggers-Stanley problem are presented.
http://arXiv.org/abs/math/0508112
http://front.math.ucdavis.edu/math.CO/0508112
(alternate)
Author(s): Alexander Gnedin and Grigori Olshanski
Abstract: The graph of zigzag diagrams is a close relative of Young's lattice. The
boundary problem for this graph amounts to describing coherent random
permutations with descent-set statistic, and is also related to certain
positive characters on the algebra of quasi-symmetric functions. We establish
connections to some further relatives of Young's lattice and solve the boundary
problem by reducing it to the classification of spreadable total orders on
integers, as recently obtained by Jacka and Warren.
http://arXiv.org/abs/math/0508131
http://front.math.ucdavis.edu/math.CO/0508131
(alternate)
Author(s): Svante Janson and Nicholas Wormald
Abstract: A rainbow subgraph of an edge-coloured graph has all edges of distinct
colours. A random d-regular graph with d even, and having edges coloured
randomly with d/2 of each of n colours, has a rainbow Hamilton cycle with
probability tending to 1 as n tends to infinity, provided d is at least 8.
http://arXiv.org/abs/math/0508145
http://front.math.ucdavis.edu/math.CO/0508145
(alternate)
Author(s): Alexis Gillett and Ronald Meester and Misja Nuyens
Abstract: We study the Bak-Sneppen model on locally finite transitive graphs $G$, in
particular on $\mathbb{Z}^d$ and on $T_{\Delta}$, the regular tree with common
degree $\Delta$. We show that the avalanches of the Bak-Sneppen model dominate
independent site percolation, in a sense to be made precise. Together with the
fact that avalanches of the Bak-Sneppen model are dominated by a simple
branching process, this yields upper and lower bounds for the critical value
$p_c^{BS}(G)$ of the Bak-Sneppen model. Our main results state that
$\frac{1}{\Delta+1} \le p_c^{BS}(T_\Delta) \le \frac{1}{\Delta -1}$, and that
$\frac{1}{2d+1}\leq p_c^{BS}(\mathbb{Z}^d)\leq \frac{1}{2d}+
\frac{1}{(2d)^2}+O\big(d^{-3}\big)$, as $d\to\infty$.
http://arXiv.org/abs/math/0508167
http://front.math.ucdavis.edu/math.PR/0508167
(alternate)
Author(s): Masayuki Kumon and Akimichi Takemura
Abstract: In the framework of the game-theoretic probability of Shafer and Vovk (2001)
it is of basic importance to construct an explicit strategy weakly forcing the
strong law of large numbers (SLLN) in the bounded forecasting game. We present
a simple finite-memory strategy based on the past average of Reality's moves,
which weakly forces the strong law of large numbers with the convergence rate
of $O(\sqrt{\log n/n})$. We also give a detailed analysis of the paths of
Skeptic's capital process for the case of the fair-coin game when our strategy
is used. We show that if Reality violates SLLN, then the exponential growth
rate of Skeptic's capital process is explicitly described in terms of the
Kullback divergence between the average of Reality's moves when she violates
SLLN and the average when she observes SLLN.
http://arXiv.org/abs/math/0508190
http://front.math.ucdavis.edu/math.PR/0508190
(alternate)
Author(s): Chris Connell and Roman Muchnik
Abstract: In this paper we extend the construction of random walks with a prescribed
Poisson boundary to the case of measures in the class of a generalized Gibbs
state. The price for dropping the $\alpha$-quasiconformal assumptions is that
we must restrict our attention to CAT($-\kappa$) groups. Apart from the new
estimates required, we prove a new approximation scheme to provide a positive
basis for positive functions in a metric measure space.
http://arXiv.org/abs/math/0507033
http://front.math.ucdavis.edu/math.GR/0507033
(alternate)
Author(s): Joseph F. Johnson
Abstract: A modification and generalisation of von Plato's fix of the frequency theory
of probability is presented. It is thermodynamic in nature. Von Plato already
fixed the logical circle in the frequency theory, we generalise his results to
not necessarily ergodic systems of classical and quantum mechanics. This turns
out to be precisely what is needed for the problem of Quantum Measurement and
the problem of induction.
http://arXiv.org/abs/quant-ph/0508059
http://front.math.ucdavis.edu/quant-ph/0508059
(alternate)
Author(s): Sourav Chatterjee
Abstract: We present a simple extension of Lindeberg's argument for the Central Limit
Theorem to get a general invariance result. We apply the technique to prove
results from random matrix theory, spin glasses, and maxima of random fields.
http://arXiv.org/abs/math/0508213
http://front.math.ucdavis.edu/math.PR/0508213
(alternate)
Author(s): Mark Conger and D. Viswanath
Abstract: Normal approximations for descents and inversions of permutations of the set
$\{1,2,...,n\}$ are well known. A number of sequences that occur in practice,
such as the human genome and other genomes, contain many repeated elements.
Motivated by such examples, we consider the number of inversions of a
permutation $\pi(1), \pi(2),...,\pi(n)$ of a multiset with $n$ elements, which
is the number of pairs $(i,j)$ with $1\leq i < j \leq n$ and $\pi(i)>\pi(j)$.
The number of descents is the number of $i$ in the range $1\leq i < n$ such
that $\pi(i) > \pi(i+1)$. We prove that, appropriately normalized, the
distribution of both inversions and descents of a random permutation of the
multiset approaches the normal distribution as $n\to\infty$, provided that the
permutation is equally likely to be any possible permutation of the multiset
and no element occurs more than $\alpha n$ times in the multiset for a fixed
$\alpha$ with $0<\alpha < 1$. Both normal approximation theorems are proved
using the size biased version of Stein's method of auxiliary randomization and
are accompanied by error bounds.
http://arXiv.org/abs/math/0508242
http://front.math.ucdavis.edu/math.PR/0508242
(alternate)
Author(s): Erkan Nanw
Abstract: We introduce a class of iterated processes called $\alpha$-time Brownian
motion for $0<\alpha \leq 2$. These are obtained by taking Brownian motion and
replacing the time parameter with a symmetric $\alpha$-stable process. We prove
a Chung-type law of the iterated logarithm (LIL) for these processes which is a
generalization of LIL proved in \cite{hu} for iterated Brownian motion. When
$\alpha =1$ it takes the following form $$ \liminf_{T\to\infty}T^{-1/2}(\log
\log T) \sup_{0\leq t\leq T}|Z_{t}|=\pi^{2}\sqrt{\lambda_{1}} a.s. $$ where
$\lambda_{1}$ is the first eigenvalue for the Cauchy process in the interval
$[-1,1].$ We also define the local time $L^{*}(x,t)$ and range $R^{*}(t)=|\{x:
Z(s)=x \text{for some} s\leq t\}|$ for these processes for $1<\alpha <2$. We
prove that there are universal constants $c_{R},c_{L}\in (0,\infty) $ such that
$$ \limsup_{t\to\infty}\frac{R^{*}(t)}{(t/\log \log t)^{1/2\alpha}\log \log t}=
c_{R} a.s. $$ $$
\liminf_{t\to\infty} \frac{\sup_{x\in \RR{R}}L^{*}(x,t)}{(t/\log \log
t)^{1-1/2\alpha}}= c_{L} a.s. $$
http://arXiv.org/abs/math/0508261
http://front.math.ucdavis.edu/math.PR/0508261
(alternate)
Author(s): Erkan nane
Abstract: We introduce a class of stochastic processes based on symmetric
$\alpha$-stable processes, for $\alpha \in (0,2]$ rational.
These are obtained by taking Markov processes and replacing the time parameter
with the modulus of a symmetric $\alpha$-stable process. We call them
$\alpha$-time processes. They generalize Brownian time processes studied in
\cite{allouba1, allouba2, allouba3}, and they introduce new interesting
examples. We establish the connection of
$\alpha-$time processes to some higher order PDE's. We also study the exit
problem for $\alpha$-time processes as they exit regular domains and connect
them to elliptic PDE's. We also obtain the PDE connection of subordinate killed
Brownian motion in bounded domains of regular boundary.
http://arXiv.org/abs/math/0508262
http://front.math.ucdavis.edu/math.PR/0508262
(alternate)
Author(s): G. Chistyakov and F. G\"otze
Abstract: We give a new approach to the definition of additive and multiplicative free
convolutions which is based on the theory of Nevanlinna and of Schur functions.
We consider the set of probability distributions as a semigroup M equipped with
the operation of free convolution and prove a Khintchine type theorem for
factorization of elements of this semigroup. Any element of M contains either
indecomposable factors or it belongs to a class, say I_0, of distributions
without indecomposable factors. In contrast to the classical convolution
semigroup in the free additive and multiplicative convolution semigroups the
class I_0 consists of units (i.e. Dirac measures) only. Furthermore we show
that the set of indecomposable elements is dense in M.
http://arXiv.org/abs/math/0508245
http://front.math.ucdavis.edu/math.OA/0508245
(alternate)
Author(s): Cosma Rohilla Shalizi and Robert Haslinger and Jean-Baptiste Rouquier and Kristina Lisa Klinkner, Cristopher Moore
Abstract: Most current methods for identifying coherent structures in
spatially-extended systems rely on prior information about the form which those
structures take. Here we present two new approaches to automatically filter the
changing configurations of spatial dynamical systems and extract coherent
structures. One, local sensitivity filtering, is a modification of the local
Lyapunov exponent approach suitable to cellular automata and other discrete
spatial systems. The other, local statistical complexity filtering, calculates
the amount of information needed for optimal prediction of the system's
behavior in the vicinity of a given point. By examining the changing
spatiotemporal distributions of these quantities, we can find the coherent
structures in a variety of pattern-forming cellular automata, without needing
to guess or postulate the form of that structure. We apply both filters to
elementary and cyclical cellular automata (ECA and CCA) and find that they
readily identify particles, domains and other more complicated structures. We
compare the results from ECA with earlier ones based upon the theory of formal
languages, and the results from CCA with a more traditional approach based on
an order parameter and free energy. While sensitivity and statistical
complexity are equally adept at uncovering structure, they are based on
different system properties (dynamical and probabilistic, respectively), and
provide complementary information.
http://arXiv.org/abs/nlin/0508001
http://front.math.ucdavis.edu/nlin.CG/0508001
(alternate)
Author(s): Itai Benjamini and Ori Gurel-Gurevich
Abstract: It is shown that the simple random walk path on a bounded degree graph,
consisting of all vertices visited and edges crossed by the walk, is almost
surely a recurrent subgraph.
http://arXiv.org/abs/math/0508270
http://front.math.ucdavis.edu/math.PR/0508270
(alternate)
Author(s): Mikhail Kovtun
Abstract: Mixed distributions are considered as a results of application of a linear
operator, which maps mixing measures to mixed measures. The main result is a
proof of continuity of this mixing operator. Corollaries for parametric
families of distributions (usually considered in literature) are also
discussed.
http://arXiv.org/abs/math/0508296
http://front.math.ucdavis.edu/math.PR/0508296
(alternate)
Author(s): Ryan O'Donnell and Michael Saks and Oded Schramm and Rocco A. Servedio
Abstract: We prove that for any decision tree calculating a boolean function
$f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n \delta_i \Inf_i(f), \]
where $\delta_i$ is the probability that the $i$th input variable is read and
$\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance,
influence and probability are taken with respect to an arbitrary product
measure on $\{-1,1\}^n$. It follows that the minimum depth of a decision tree
calculating a given balanced function is at least the reciprocal of the largest
influence of any input variable. Likewise, any balanced boolean function with a
decision tree of depth $d$ has a variable with influence at least
$\frac{1}{d}$. The only previous nontrivial lower bound known was $\Omega(d
2^{-d})$. Our inequality has many generalizations, allowing us to prove
influence lower bounds for randomized decision trees, decision trees on
arbitrary product probability spaces, and decision trees with non-boolean
outputs. As an application of our results we give a very easy proof that the
randomized query complexity of nontrivial monotone graph properties is at least
$\Omega(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p \leq
\half$ is the critical threshold probability. This supersedes the milestone
$\Omega(v^{4/3})$ bound of Hajnal and is sometimes superior to the best known
lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.
http://arXiv.org/abs/cs/0508071
http://front.math.ucdavis.edu/cs.CC/0508071
(alternate)
Author(s): Ronald Ortner
Abstract: We show that combinations of optimal (stationary) policies in unichain Markov
decision processes are optimal. That is, let M be a unichain Markov decision
process with state space S, action space A and policies \pi_j^*: S -> A (1\leq
j\leq n) with optimal average infinite horizon reward. Then any combination \pi
of these policies, where for each state i in S there is a j such that
\pi(i)=\pi_j^*(i), is optimal as well. Furthermore, we prove that any mixture
of optimal policies, where at each visit in a state i an arbitrary action
\pi_j^*(i) of an optimal policy is chosen, yields optimal average reward, too.
http://arXiv.org/abs/math/0508319
http://front.math.ucdavis.edu/math.CO/0508319
(alternate)
Author(s): Gopal K. Basak and Amites Dasgupta
Abstract: We take a unified approach to central limit theorems for a class of
irreducible urn models with constant replacement matrix. Depending on the
eigenvalue, we consider appropriate linear combinations of the number of balls
of different colors. Then under appropriate norming the multivariate
distribution of the weak limits of these linear combinations is obtained and
independence and dependence issues are investigated.
http://arXiv.org/abs/math/0507084
http://front.math.ucdavis.edu/math.PR/0507084
(alternate)
Author(s): Teresa Mart\'inez and Marta Sanz-Sol\'e
Abstract: We study a stochastic boundary value problem on $(0,1)^d$ of elliptic type in
dimension $d\ge 4$, driven by a coloured noise. An approximation scheme based
on a suitable discretization of the Laplacian on a lattice of $(0,1)^d$ is
presented; we also give the rate of convergence to the original SPDE in
$L^p(\Omega;L^{2}(D))$--norm, for some values of $p$.
http://arXiv.org/abs/math/0508339
http://front.math.ucdavis.edu/math.PR/0508339
(alternate)
Author(s): Gady Kozma
Abstract: We show that the scaling limit exists and is invariant to dilations and
rotations. We give some tools that might be useful to show universality.
http://arXiv.org/abs/math/0508344
http://front.math.ucdavis.edu/math.PR/0508344
(alternate)
Author(s): Klaus Fleischmann and Peter Moerters and and Vitali Wachtel
Abstract: We consider the behaviour of a continuous super-Brownian motion catalysed by
a random medium with infinite overall density under the hydrodynamic scaling of
mass, time, and space. We show that, in supercritical dimensions, the scaled
process converges to a macroscopic heat flow, and the appropriately rescaled
random fluctuations around this macroscopic flow are asymptotically bounded, in
the sense of log-Laplace transforms, by generalised stable Ornstein-Uhlenbeck
processes. The most interesting new effect we observe is the occurrence of an
index-jump from a 'Gaussian' situation to stable fluctuations of index 1+gamma,
where gamma is an index associated to the medium.
http://arXiv.org/abs/math/0508368
http://front.math.ucdavis.edu/math.PR/0508368
(alternate)
Author(s): Saul Jacka and Jon Warren
Abstract: In this paper we study random orderings of the integers with a certain
invariance property. We describe all such orders in a simple way. We define and
represent random shuffles of a countable set of labels and then give an
interpretation of these orders in terms of a class of generalized riffle
shuffles.
http://arXiv.org/abs/math/0508369
http://front.math.ucdavis.edu/math.PR/0508369
(alternate)
Author(s): Gregory Berkolaiko and Alexandra Rodkina
Abstract: We consider a non-homogeneous nonlinear stochastic difference equation
X_{n+1} = X_n (1 + f(X_n)\xi_{n+1}) + S_n, and its important special case
X_{n+1} = X_n (1 + \xi_{n+1}) + S_n, both with initial value X_0, non-random
decaying free coefficient S_n and independent random variables \xi_n. We
establish results on \as convergence of solutions X_n to zero. The necessary
conditions we find tie together certain moments of the noise \xi_n and the rate
of decay of S_n. To ascertain sharpness of our conditions we discuss some
situations when X_n diverges. We also establish a result concerning the rate of
decay of X_n to zero.
http://arXiv.org/abs/math/0508371
http://front.math.ucdavis.edu/math.PR/0508371
(alternate)
Author(s): T.V. Dudnikova and A.I. Komech and N.E. Ratanov and Yu.M. Suhov
Abstract: The paper considers the wave equation, with constant or variable coefficients
in $\R^n$, with odd $n\geq 3$. We study the asymptotics of the distribution
$\mu_t$ of the random solution at time $t\in\R$ as $t\to\infty$. It is assumed
that the initial measure $\mu_0$ has zero mean, translation-invariant
covariance matrices, and finite expected energy density. We also assume that
$\mu_0$ satisfies a Rosenblatt- or Ibragimov-Linnik-type space mixing
condition. The main result is the convergence of $\mu_t$ to a Gaussian measure
$\mu_\infty$ as $t\to\infty$, which gives a Central Limit Theorem (CLT) for the
wave equation. The proof for the case of constant coefficients is based on an
analysis of long-time asymptotics of the solution in the Fourier representation
and Bernstein's `room-corridor' argument. The case of variable coefficients is
treated by using a version of the scattering theory for infinite energy
solutions, based on Vainberg's results on local energy decay.
http://arXiv.org/abs/math-ph/0508039
http://front.math.ucdavis.edu/math-ph/0508039
(alternate)
Author(s): E. Ben-Naim and P.L. Krapivsky
Abstract: We present a statistical analysis of biological evolution processes.
Specifically, we study the stochastic replication-mutation-death model where
the population of a species may grow or shrink by birth or death, respectively,
and additionally, mutations lead to the creation of new species. We rank the
various species by the chronological order by which they originate. The average
population N_k of the kth species decays algebraically with rank, N_k ~ M^{mu}
k^{-mu}, where M is the average total population. The characteristic exponent
mu=(alpha-gamma)/(alpha+beta-gamma)$ depends on alpha, beta, and gamma, the
replication, mutation, and death rates. Furthermore, the average population P_k
of all descendants of the kth species has a universal algebraic behavior, P_k ~
M/k.
http://arXiv.org/abs/q-bio/0508023
http://front.math.ucdavis.edu/q-bio.PE/0508023
(alternate)
Author(s): Wlodzimierz Bryc and Jacek Wesolowski
Abstract: We give an elementary construction of a time-invertible Markov process which
is discrete except at one instance. The process is one of the quadratic
harnesses studied in our previous papers and can be regarded as a random joint
of two independent Poisson processes.
http://arXiv.org/abs/math/0508383
http://front.math.ucdavis.edu/math.PR/0508383
(alternate)
Author(s): M. Draief and A. Ganesh
Abstract: In recent work, Jon Kleinberg considered a small-world network model
consisting of a d-dimensional lattice augmented with shortcuts. The probability
of a shortcut being present between two points decays as a power of the
distance between them. Kleinberg studied the efficiency of greedy routing
depending on the value of the power. The results were extended to a continuum
model by Franceschetti and Meester. In our work, we extend the result to more
realistic models constructed from a Poisson point process, wherein each point
is connected to all its neighbours within some fixed radius, as well as
possessing random shortcuts to more distant nodes as described above.
http://arXiv.org/abs/math/0508410
http://front.math.ucdavis.edu/math.PR/0508410
(alternate)
Author(s): Boris Tsirelson
Abstract: We compare two examples of random dense countable sets, `Brownian local
minima' and `unordered uniform infinite sample'. They appear to be identically
distributed. A framework for such notions is proposed. In addition, random
elements of other singular spaces (especially, reals modulo rationals) are
considered.
http://arXiv.org/abs/math/0508414
http://front.math.ucdavis.edu/math.PR/0508414
(alternate)
Author(s): Djalil Chafai (LSProba and Upte Umr Inra/Envt 181) and Didier Concordet (LSProba, Upte Umr Inra/Envt 181)
Abstract: The aim of this article is to provide a strong consistency Theorem for
approximated M-estimators. It contains both Wald and Pfanzagl type results for
maximum likelihood. The proof relies, in particular, on the existence of a sort
of contraction of the parameter space which admits the true parameter as a
fixed point. In a way, it can be seen as a simplification of ideas of Wang and
Pfanzagl, generalised to approximated M-estimators. Proofs are short and
elementary.
http://arXiv.org/abs/math/0507102
http://front.math.ucdavis.edu/math.PR/0507102
(alternate)
Author(s): T.V. Dudnikova and A.I. Komech and E.A. Kopylova and Yu.M. Suhov
Abstract: Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant
or variable coefficients. We study the distribution $\mu_t$ of the random
solution at time $t\in\R$. We assume that the initial probability measure
$\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean
energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or
Ibragimov-Linnik-type mixing condition. The main result is the convergence of
$\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central
Limit Theorem for the KGE. The proof for the case of constant coefficients is
based on an analysis of long time asymptotics of the solution in the Fourier
representation and Bernstein's `room-corridor' argument. The case of variable
coefficients is treated by using an `averaged' version of the scattering theory
for infinite energy solutions, based on Vainberg's results on local energy
decay.
http://arXiv.org/abs/math-ph/0508042
http://front.math.ucdavis.edu/math-ph/0508042
(alternate)
Author(s): T.V. Dudnikova and A.I. Komech and H. Spohn
Abstract: Consider the wave equation with constant or variable coefficients in $\R^3$.
The initial datum is a random function with a finite mean density of energy
that also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition.
The random function converges to different space-homogeneous processes as
$x_3\to\pm\infty$, with the distributions $\mu_\pm$. We study the distribution
$\mu_t$ of the random solution at a time $t\in\R$. The main result is the
convergence of $\mu_t$ to a Gaussian translation-invariant measure as
$t\to\infty$ that means central limit theorem for the wave equation. The proof
is based on the Bernstein `room-corridor' argument. The application to the case
of the Gibbs measures $\mu_\pm=g_\pm$ with two different temperatures $T_{\pm}$
is given. Limiting mean energy current density formally is $-\infty\cdot
(0,0,T_+ -T_-)$ for the Gibbs measures, and it is finite and equals to
$-C(0,0,T_+ -T_-)$ with $C>0$ for the convolution with a nontrivial test
function.
http://arXiv.org/abs/math-ph/0508044
http://front.math.ucdavis.edu/math-ph/0508044
(alternate)
Author(s): Le Anh Vinh
Abstract: In this paper we study random walks on the hypergroup of circles in a finite
field of prime order p = 4l + 3. We investigating the behavior of random walks
on this hypergroup, the equilibrium distribution and the mixing times. We use
two different approaches - comparision of Dirichlet forms (geometric bound of
eigenvalues), and coupling methods, to show that the mixing time of random
walks on hypergroup of circles is only linear.
http://arXiv.org/abs/math/0508403
http://front.math.ucdavis.edu/math.CO/0508403
(alternate)
Author(s): Tai Melcher
Abstract: Let G be a Lie group equipped with a set of left invariant vector fields.
These vector fields generate a function \xi on Wiener space into G via the
stochastic version of Cartan's rolling map. It is shown here that, for any
smooth function f with compact support, f(\xi) is Malliavin differentiable to
all orders and these derivatives belong to L^p(\mu) for all p>1, where \mu is
Wiener measure.
http://arXiv.org/abs/math/0508419
http://front.math.ucdavis.edu/math.PR/0508419
(alternate)
Author(s): Martin Hildebrand
Abstract: Chung, Diaconis, and Graham considered random processes of the form
X_{n+1}=2X_n+b_n (mod p) where X_0=0, p is odd, and b_n for n=0,1,2,... are
i.i.d. random variables on {-1,0,1}. If Pr(b_n=-1)= Pr(b_n=1)=\beta and
Pr(b_n=0)=1-2\beta, they asked which value of \beta makes X_n get close to
uniformly distributed on the integers mod p the slowest. In this paper, we
extend the results of Chung, Diaconis, and Graham in the case p=2^t-1 to show
that for 0<\beta<=1/2, there is no such value of \beta.
http://arXiv.org/abs/math/0508427
http://front.math.ucdavis.edu/math.PR/0508427
(alternate)
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan
Abstract: Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of
intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$
within distance $r$ independently with probability $p$. This is the simplest
case of a `spread-out' percolation model studied by Penrose, who showed that,
as $r\to\infty$, the average degree of the corresponding random graph at the
percolation threshold tends to 1, i.e., the percolation threshold and the
threshold for criticality of the naturally associated branching process
approach one another. Here we show that this result follows immediately from of
a general result of the authors on inhomogeneous random graphs.
http://arXiv.org/abs/math/0508430
http://front.math.ucdavis.edu/math.PR/0508430
(alternate)
Author(s): Tai Melcher
Abstract: This paper discusses the existence of gradient estimates for second order
hypoelliptic heat kernels on manifolds. It is now standard that such
inequalities, in the elliptic case, are equivalent to a lower bound on the
Ricci tensor of the Riemannian metric. For hypoelliptic operators, the
associated ``Ricci curvature'' takes on the value -\infty at points of
degeneracy of the semi-Riemannian metric associated to the operator. For this
reason, the standard proofs for the elliptic theory fail in the hypoelliptic
setting.
This paper presents recent results for hypoelliptic operators. Malliavin
calculus methods transfer the problem to one of determining certain infinite
dimensional estimates. Here, the underlying manifold is a Lie group, and the
hypoelliptic operators are invariant under left translation. In particular,
``L^p-type'' gradient estimates hold for p\in(1,\infty), and the p=2 gradient
estimate implies a Poincar\'e estimate in this context.
http://arXiv.org/abs/math/0508420
http://front.math.ucdavis.edu/math.AP/0508420
(alternate)
Author(s): Jozsef Balogh and Robin Pemantle
Abstract: An infinite sequence of 0's and 1's evolves by flipping each~1 to a~0
exponentially at rate one. When a~1 flips, all bits to its right also flip.
Starting from any configuration with finitely many 1's to the left of the
origin, we show that the leftmost~1 moves right with linear speed. Upper and
lower bounds are given on the speed.
http://arXiv.org/abs/math/0506626
http://front.math.ucdavis.edu/math.PR/0506626
(alternate)
Author(s): P.Okunev
Abstract: We propose a fast algorithm for computing the economic capital, Value at Risk
and Greeks in the Gaussian factor model. The algorithm proposed here is much
faster than brute force Monte Carlo simulations or Fourier transform based
methods \cite{MD}. While the algorithm of Hull-White \cite{HW} is comparably
fast, it assumes that all the loans in the portfolio have equal notionals and
recovery rates. This is a very restrictive assumption which is unrealistic for
many portfolios encountered in practice. Our algorithm makes no assumptions
about the homogeneity of the portfolio. Additionally, it is easier to implement
than the algorithm of Hull-White. We use the implicit function theorem to
derive analytic expressions for the Greeks.
http://arXiv.org/abs/math/0507082
http://front.math.ucdavis.edu/math.ST/0507082
(alternate)
Author(s): P.Chigansky
Abstract: The filtering problem for a finite state Markov chain observed in white noise
is addressed in continuous time. The low signal to noise asymptotic is derived
for the performance indices of MAP and MMSE estimates of the signal.
http://arXiv.org/abs/math/0508446
http://front.math.ucdavis.edu/math.PR/0508446
(alternate)
Author(s): Javier Parcet
Abstract: Given a probability space $(\Omega, \mathsf{A}, \mu)$, let $\mathsf{A}_1,
\mathsf{A}_2, ...$ be a filtration of $\sigma$-subalgebras of $\mathsf{A}$ and
let $\mathsf{E}_1, \mathsf{E}_2, ...$ denote the corresponding family of
conditional expectations. Given a martingale $f = (f_1, f_2, ...)$ adapted to
this filtration and bounded in $L_p(\Omega)$ for some $2 \le p < \infty$,
Burkholder's inequality claims that $$\|f\|_{L_p(\Omega)} \sim_{\mathrm{c}_p}
\Big\| \Big(\sum_{k=1}^\infty \mathsf{E}_{k-1}(|df_k|^2) \Big)^{1/2}
\Big\|_{L_{p}(\Omega)} + \Big(\sum_{k=1}^\infty \|df_k\|_p^p \Big)^{1/p}.$$
Motivated by quantum probability, Junge and Xu recently extended this result to
the range $1 < p < 2$. In this paper we study Burkholder's inequality for
$p=1$, for which the techniques (as we shall explain) must be different. Quite
surprisingly, we obtain two non-equivalent estimates which play the role of the
weak type $(1,1)$ analog of Burkholder's inequality. As application, we obtain
new properties of Davis decomposition for martingales.
http://arXiv.org/abs/math/0508447
http://front.math.ucdavis.edu/math.PR/0508447
(alternate)
Author(s): Ying Hu and Peter Imkeller and Matthias Muller
Abstract: We consider the problem of utility maximization for small traders on
incomplete financial markets. As opposed to most of the papers dealing with
this subject, the investors' trading strategies we allow underly constraints
described by closed, but not necessarily convex, sets. The final wealths
obtained by trading under these constraints are identified as stochastic
processes which usually are supermartingales, and even martingales for
particular strategies. These strategies are seen to be optimal, and the
corresponding value functions determined simply by the initial values of the
supermartingales. We separately treat the cases of exponential, power and
logarithmic utility.
http://arXiv.org/abs/math/0508448
http://front.math.ucdavis.edu/math.PR/0508448
(alternate)
Author(s): Patrick Cheridito and Damir Filipovic and Marc Yor
Abstract: We provide explicit sufficient conditions for absolute continuity and
equivalence between the distributions of two jump-diffusion processes that can
explode and be killed by a potential.
http://arXiv.org/abs/math/0508450
http://front.math.ucdavis.edu/math.PR/0508450
(alternate)
Author(s): Malwina J. Luczak and Colin McDiarmid
Abstract: Suppose that there are n bins, and balls arrive in a Poisson process at rate
\lambda n, where \lambda >0 is a constant. Upon arrival, each ball chooses a
fixed number d of random bins, and is placed into one with least load. Balls
have independent exponential lifetimes with unit mean. We show that the system
converges rapidly to its equilibrium distribution; and when d\geq 2, there is
an integer-valued function m_d(n)=\ln \ln n/\ln d+O(1) such that, in the
equilibrium distribution, the maximum load of a bin is concentrated on the two
values m_d(n) and m_d(n)-1, with probability tending to 1, as n\to \infty. We
show also that the maximum load usually does not vary by more than a constant
amount from \ln \ln n/\ln d, even over quite long periods of time.
http://arXiv.org/abs/math/0508451
http://front.math.ucdavis.edu/math.PR/0508451
(alternate)
Author(s): Fabrice Baudoin and Josef Teichmann
Abstract: We apply methods from Malliavin calculus to prove an infinite-dimensional
version of Hormander's theorem for stochastic evolution equations in the spirit
of Da Prato-Zabczyk. This result is used to show that HJM-equations from
interest rate theory, which satisfy the Hormander condition, have the
conceptually undesirable feature that any selection of yields admits a density
as multi-dimensional random variable.
http://arXiv.org/abs/math/0508452
http://front.math.ucdavis.edu/math.PR/0508452
(alternate)
Author(s): Serik Sagitov and Peter Jagers
Abstract: We establish convergence to the Kingman coalescent for a class of
age-structured population models with time-constant population size. Time is
discrete with unit called a year. Offspring numbers in a year may depend on
mother's age.
http://arXiv.org/abs/math/0508454
http://front.math.ucdavis.edu/math.PR/0508454
(alternate)
Author(s): Jianfeng Zhang
Abstract: In this paper we investigate a class of decoupled forward-backward SDEs,
where the volatility of the FSDE is degenerate and the terminal value of the
BSDE is a discontinuous function of the FSDE. Such an FBSDE is associated with
a degenerate parabolic PDE with discontinuous terminal condition. We first
establish a Feynman-Kac type representation formula for the spatial derivative
of the solution to the PDE. As a consequence, we show that there exists a
stopping time \tau such that the martingale integrand of the BSDE is continuous
before \tau and vanishes after \tau. However, it may blow up at \tau, as
illustrated by an example. Moreover, some estimates for the martingale
integrand before \tau are obtained. These results are potentially useful for
pricing and hedging discontinuous exotic options (e.g., digital options) when
the underlying asset's volatility is small, and they are also useful for
studying the rate of convergence of finite-difference approximations for
degenerate parabolic PDEs.
http://arXiv.org/abs/math/0508457
http://front.math.ucdavis.edu/math.PR/0508457
(alternate)
Author(s): G. J. Morrow and Y. Zhang
Abstract: Let L_n denote the lowest crossing of a square 2n\times2n box for critical
site percolation on the triangular lattice imbedded in Z^2. Denote also by F_n
the pioneering sites extending below this crossing, and Q_n the pivotal sites
on this crossing. Combining the recent results of Smirnov and Werner [Math.
Res. Lett. 8 (2001) 729-744] on asymptotic probabilities of multiple arm paths
in both the plane and half-plane, Kesten's [Comm. Math. Phys. 109 (1987)
109-156] method for showing that certain restricted multiple arm paths are
probabilistically equivalent to unrestricted ones, and our own second and
higher moment upper bounds, we obtain the following results. For each positive
integer \tau, as n\to\infty: 1. E(|L_n|^{\tau})=n^{4\tau/3+o(1)}. 2.
E(|F_n|^{\tau})=n^{7\tau/4+o(1)}. 3. E(|Q_n|^{\tau})=n^{3\tau/4+o(1)}. These
results extend to higher moments a discrete analogue of the recent results of
Lawler, Schramm and Werner [Math. Res. Lett. 8 (2001) 401-411] that the
frontier, pioneering points and cut points of planar Brownian motion have
Hausdorff dimensions, respectively, 4/3, 7/4 and 3/4.
http://arXiv.org/abs/math/0508459
http://front.math.ucdavis.edu/math.PR/0508459
(alternate)
Author(s): Amarjit Budhiraja and Arka Prasanna Ghosh
Abstract: In this work we study the problem of asymptotically optimal control of a
well-known multi-class queuing network, referred to as the ``crisscross
network,'' in heavy traffic. We consider exponential inter-arrival and service
times, linear holding cost and an infinite horizon discounted cost criterion.
In a suitable parameter regime, this problem has been studied in detail by
Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996) 2133-2171] using
viscosity solution methods. In this work, using the pathwise solution of the
Brownian control problem, we present an elementary and transparent treatment of
the problem (with the identical parameter regime as in [SIAM J. Control Optim.
34 (1996) 2133-2171]) using large deviation ideas introduced in [Ann. Appl.
Probab. 10 (2000) 75-103, Ann. Appl. Probab. 11 (2001) 608-649]. We obtain an
asymptotically optimal scheduling policy which is of threshold type. The proof
is of independent interest since it is one of the few results which gives the
asymptotic optimality of a control policy for a network with a more than
one-dimensional workload process.
http://arXiv.org/abs/math/0508460
http://front.math.ucdavis.edu/math.PR/0508460
(alternate)
Author(s): Serguei Foss and Zbigniew Palmowski and Stan Zachary
Abstract: We study the asymptotic probability that a random walk with heavy-tailed
increments crosses a high boundary on a random time interval. We use new
techniques to extend results of Asmussen [Ann. Appl. Probab. 8 (1998) 354-374]
to completely general stopping times, uniformity of convergence over all
stopping times and a wide class of nonlinear boundaries. We also give some
examples and counterexamples.
http://arXiv.org/abs/math/0508461
http://front.math.ucdavis.edu/math.PR/0508461
(alternate)
Author(s): Benedicte Haas
Abstract: This paper introduces stochastic processes that describe the evolution of
systems of particles in which particles immigrate according to a Poisson
measure and split according to a self-similar fragmentation. Criteria for
existence and absence of stationary distributions are established and
uniqueness is proved. Also, convergence rates to the stationary distribution
are given. Linear equations which are the deterministic counterparts of
fragmentation with immigration processes are next considered. As in the
stochastic case, existence and uniqueness of solutions, as well as existence
and uniqueness of stationary solutions, are investigated.
http://arXiv.org/abs/math/0508462
http://front.math.ucdavis.edu/math.PR/0508462
(alternate)
Author(s): Mathew D. Penrose
Abstract: Given $n$ independent random marked $d$-vectors $X_i$ with a common density,
define the measure $\nu_n = \sum_i \xi_i $, where $\xi_i$ is a measure (not
necessarily a point measure) determined by the (suitably rescaled) set of
points near $X_i$. Technically, this means here that $\xi_i$ stabilizes with a
suitable power-law decay of the tail of the radius of stabilization. For
bounded test functions $f$ on $R^d$, we give a law of large numbers and central
limit theorem for $\nu_n(f)$. The latter implies weak convergence of
$\nu_n(\cdot)$, suitably scaled and centred, to a Gaussian field acting on
bounded test functions. The general result is illustrated with applications
including the volume and surface measure of germ-grain models with unbounded
grain sizes.
http://arXiv.org/abs/math/0508464
http://front.math.ucdavis.edu/math.PR/0508464
(alternate)
Author(s): Svante Janson and Malwina Luczak
Abstract: We study the k-core of a random (multi)graph on n vertices with a given
degree sequence. We let n tend to infinity. Then, under some regularity
conditions on the degree sequences, we give conditions on the asymptotic shape
of the degree sequence that imply that with high probability the k-core is
empty, and other conditions that imply that with high probability the k-core is
non-empty and the sizes of its vertex and edge sets satisfy a law of large
numbers; under suitable assumptions these are the only two possibilities. In
particular, we recover the result by Pittel, Spencer and Wormald on the
existence and size of a k-core in G(n,p) and G(n,m).
Our method is based on the properties of empirical distributions of
independent random variables, and leads to simple proofs.
http://arXiv.org/abs/math/0508453
http://front.math.ucdavis.edu/math.CO/0508453
(alternate)
Author(s): Sourav Chatterjee
Abstract: In this article, we present a general technique for analyzing the
concentration of Haar measures on compact groups using the properties of
certain kinds of random walks. As an application, we obtain a new kind of
measure concentration for random unitary matrices, which allows us to directly
establish the concentration of the empirical distribution of eigenvalues of a
class of random matrices. The end-result of this application is a quantitative
version of Voiculescu's celebrated connection between random matrices and free
probability.
http://arXiv.org/abs/math/0508518
http://front.math.ucdavis.edu/math.PR/0508518
(alternate)
Author(s): Sourav Chatterjee
Abstract: We present a generalization of Lindeberg's method of proving the central
limit theorem to encompass general smooth functions (instead of just sums) and
dependent random variables. The technique is then used to obtain an invariance
result for smooth functions of exchangeable random variables. As an
illustrative application of this theorem, we then establish ``convergence to
Wigner's law'' for eigenspectra of matrices with exchangeable random entries.
http://arXiv.org/abs/math/0508519
http://front.math.ucdavis.edu/math.PR/0508519
(alternate)
Author(s): Anatoli Manita and Francois Simonot
Abstract: We consider a cascade model of $N$ different processors performing a
distributed parallel simulation. The main goal of the study is to show that the
long-time dynamics of the system has a cluster behavior. To attack this problem
we combine two methods: stochastic comparison and Foster-Lyapunov functions.
http://arXiv.org/abs/math/0508533
http://front.math.ucdavis.edu/math.PR/0508533
(alternate)
Author(s): Vladislav Kargin
Abstract: An analogue of the Bernstein inequality is derived for partial sums of a
vector-valued function on a finite reversible Markov chain. The inequality
gives an upper bound for the probability of a large deviation of the partial
sum. The bound depends on the chain's spectral gap, the dimension of the space
where the function takes values, and the upper bound on the size and the
variance of the function.
http://arXiv.org/abs/math/0508538
http://front.math.ucdavis.edu/math.PR/0508538
(alternate)
Author(s): T.V. Dudnikova and A.I. Komech
Abstract: We consider the dynamics of a field coupled to a harmonic crystal with $n$
components in dimension $d$, $d,n\ge 1$. The crystal and the dynamics are
translation-invariant with respect to the subgroup $\Z^d$ of $\R^d$. The
initial data is a random function with a finite mean density of energy which
also satisfies a Rosenblatt- or Ibragimov-Linnik-type mixing condition.
Moreover, initial correlation functions are translation-invariant with respect
to the discrete subgroup $\Z^d$. We study the distribution $\mu_t$ of the
solution at time $t\in\R$. The main result is the convergence of $\mu_t$ to a
Gaussian measure as $t\to\infty$, where $\mu_\infty$ is translation-invariant
with respect to the subgroup $\Z^d$.
http://arXiv.org/abs/math-ph/0508053
http://front.math.ucdavis.edu/math-ph/0508053
(alternate)
Author(s): Matyas Barczy and Gyula Pap
Abstract: First we give a construction of bridges derived from a general Markov process
using only its transition densities. We give sufficient conditions for their
existence and uniqueness (in law). Then we prove that the law of the radial
part of the bridge with endpoints zero derived from a special multidimensional
Ornstein-Uhlenbeck process equals the law of the bridge with endpoints zero
derived from the radial part of the same Ornstein-Uhlenbeck process. We also
construct bridges derived from general multidimensional Ornstein-Uhlenbeck
processes.
http://arXiv.org/abs/math/0508542
http://front.math.ucdavis.edu/math.PR/0508542
(alternate)
Author(s): Amir Dembo and Nina Gantert and Yuval Peres and Zhan Shi
Abstract: Let $\xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional
random walk in random environment after $n$ steps, and consider the maximum
$\xi^*(n) = \max_x \xi(n,x)$. It is known that $\limsup \xi^*(n)/n$ is a
positive constant a.s. We prove that $\liminf_n (\log\log\log n)\xi^*(n)/n$ is
a positive constant a.s.; this answers a question of P. R\'ev\'esz (1990). The
proof is based on an analysis of the {\em valleys /} in the environment,
defined as the potential wells of record depth. In particular, we show that
almost surely, at any time $n$ large enough, the random walker has spent almost
all of its lifetime in the two deepest valleys of the environment it has
encountered. We also prove a uniform exponential tail bound for the ratio of
the expected total occupation time of a valley and the expected local time at
its bottom.
http://arXiv.org/abs/math/0508579
http://front.math.ucdavis.edu/math.PR/0508579
(alternate)
Author(s): Yuval Peres and Oded Schramm and Scott Sheffield and David B. Wilson
Abstract: The game of Hex has two players who take turns placing stones of their colors
on the hexagons of a rhombus-shaped hexagonal grid. Black wins by completing a
crossing between two opposite edges, while White wins by completing a crossing
between the other pair of opposite edges. Although ordinary Hex is famously
difficult to analyze, random-turn Hex--in which players toss a coin before each
turn to decide who gets to place the next stone--has a simple optimal strategy.
It belongs to a general class of random-turn games--called selection games--in
which the expected payoff when both players play the random-turn game optimally
is the same as when both players play randomly. We also describe the optimal
strategy and study the expected length of the game under optimal play for
random-turn Hex and several other selection games.
http://arXiv.org/abs/math/0508580
http://front.math.ucdavis.edu/math.PR/0508580
(alternate)
Author(s): Piotr Rozmej and Anna Karczewska
Abstract: In the paper we study some numerical solutions to Volterra equations which
interpolate heat and wave equations. We present a scheme for construction of
approximate numerical solutions for one and two spatial dimensions. Some
solutions to the stochastic version of such equations (for one spatial
dimension) are presented as well.
http://arXiv.org/abs/math/0508564
http://front.math.ucdavis.edu/math.NA/0508564
(alternate)
Author(s): Francis Comets (PMA) and Francois Delarue (PMA) and Rene Schott (IEC and LORIA)
Abstract: We provide a probabilistic analysis of the banker algorithm when transition
probabilities may depend on time and space. The transition probabilities
evolve, as time goes by, along the trajectory of an ergodic Markovian
environment, whereas the spatial parameter just acts on long runs. Our model
appears as a new (small) step towards more general time and space dependent
protocols. Our analysis relies on well-known results in stochastic
homogenization theory and investigates the asymptotic behaviour of the rescaled
algorithm as the total amount of resource available for allocation tends to the
infinity. In the two dimensional setting, we manage to exhibit three different
possible regimes for the deadlock time of the limit system.
http://arXiv.org/abs/math/0507115
http://front.math.ucdavis.edu/math.PR/0507115
(alternate)
Author(s): Francis Comets (PMA) and Serguei Popov (IME)
Abstract: We study branching random walks in random i.i.d. environment in $\Z^d, d \geq
1$. For this model, the population size cannot decrease, and a natural
definition of recurrence is introduced. We prove a dichotomy for
recurrence/transience, depending only on the support of the environmental law.
We give sufficient conditions for recurrence and for transience. In the
recurrent case, we study the asymptotics of the tail of the distribution of the
hitting times and prove a shape theorem for the set of lattice sites which are
visited up to a large time.
http://arXiv.org/abs/math/0507126
http://front.math.ucdavis.edu/math.PR/0507126
(alternate)
Author(s): Olivier Garet (MAPMO) and R\'{e}gine Marchand (IEC)
Abstract: We study a competition model on $\mathbb{Z}^d$ where the two infections are
driven by supercritical Bernoulli percolations with distinct parameters $p$ and
$q$. We prove that, for any $q$, there exist at most countably many values of
$p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.
http://arXiv.org/abs/math/0507133
http://front.math.ucdavis.edu/math.PR/0507133
(alternate)
Author(s): Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (BR-IMPA) and Herbert Spohn (D-MUTU-ZM), Eulalia Vares (BR-CBPF)
Abstract: In this article we discuss a set of geometric ideas which shed some light on
the question of directed polymer pinning in the presence of bulk disorder.
Differing from standard methods and techniques, we transform the problem to a
particular dependent percolative system and relate the pinning transition to a
percolation transition.
http://arXiv.org/abs/math/0507142
http://front.math.ucdavis.edu/math.PR/0507142
(alternate)
Author(s): Aureli Alabert and Marco Ferrante
Abstract: We consider linear stochastic differential-algebraic equations with constant
coefficients and additive white noise. Due to the nature of this class of
equations, the solution must be defined as a generalised process (in the sense
of Dawson and Fernique). We provide sufficient conditions for the law of the
variables of the solution process to be absolutely continuous with respect to
Lebesgue measure.
http://arXiv.org/abs/math/0507159
http://front.math.ucdavis.edu/math.PR/0507159
(alternate)
Author(s): Daniel Commenges and Anne Gegout-Petit
Abstract: We define a general coarsening model for stochastic processes. We decribe
incomplete data by means of sigma-fields and we give conditions of ignorability
for likelihood inference.
http://arXiv.org/abs/math/0507151
http://front.math.ucdavis.edu/math.ST/0507151
(alternate)
Author(s): W. Hachem and P. Loubaton and J. Najim
Abstract: Consider a $N\times n$ random matrix $ Y_n$ where the entries are independent
but not identically distributed (matrices with a variance profile) Consider now
a deterministic $N\times n$ matrix $A_n$ whose columns and rows are uniformly
bounded for the Euclidean norm.
Let $\Sigma_n=Y_n+A_n$. We prove in this article that there exists a
deterministic equivalent to the empirical Stieltjes transform of the
distribution of the eigenvalues of $\Sigma_n \Sigma_n^T$ which is itself the
Stieltjes transform of a probability measure.
This work is motivated by the context of performance evaluation of Multiple
Inputs / Multiple Output (MIMO) wireless digital communication channels. As an
application, we derive a deterministic equivalent to the mutual information of
a wireless channel.
http://arXiv.org/abs/math/0507172
http://front.math.ucdavis.edu/math.PR/0507172
(alternate)
Author(s): Francesco Caravenna and Giambattista Giacomin and Lorenzo Zambotti
Abstract: We consider a general model of an heterogeneous polymer chain fluctuating in
the proximity of an interface between two selective solvents. The heterogeneous
character of the model comes from the fact that monomer units interact with the
solvents and with the interface according to some charges that they carry. The
charges repeat themselves along the chain in a periodic fashion. The main
question on this model is whether the polymer remains tightly close to the
interface, a phenomenon called localization, or there is a marked preference
for one of the two solvents yielding thus a delocalization phenomenon.
We propose an approach to this model, based on renewal theory, that yields
sharp estimates on the partition function of the model in all the regimes
(localized, delocalized and critical). This in turn allows to get a very
precise description of the polymer measure, both in a local sense
(thermodynamic limit) and in a global sense (scaling limits). A key point, but
also a byproduct, of our analysis is the closeness of the polymer measure to
suitable Markov Renewal Processes.
http://arXiv.org/abs/math/0507178
http://front.math.ucdavis.edu/math.PR/0507178
(alternate)
Author(s): Bernard Roynette and Pierre Vallois and Agnes Volpi
Abstract: Let (X_t, t>=0) be a Levy process started at 0, with Levy measure nu and T_x
the first hitting time of level x>0: T_x:=inf{t>=0; X_t>x}. Let $F(theta, mu,
rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x)
:=E(e^(-theta T_x - mu K_x \rho L_x) 1_(T_x<+infinity)), where theta>=0, mu>=0,
rho>=0, x>=0, K_x:=X_(T_x)-x and L_x:=x-X_(T_(x^-)). If we assume that nu has
finite exponential moments we exhibit an asymptotic expansion for
F(theta,mu,rho,x), as x -> +infinity. A limit theorem involving a normalization
of the triplet (T_x,K_x,L_x) as x -> +infinity, may be deduced. At last, if
nu_(|_R_+) has finite moment of fixed order, we prove that the ruin probability
P(T_x<+infinity) has at most a polynomial decay.
http://arXiv.org/abs/math/0507193
http://front.math.ucdavis.edu/math.PR/0507193
(alternate)
Author(s): Stefan Cobza\c{s}
Abstract: The aim of the present paper is to prove that the family of all closed
nonempty subsets of a complete probabilistic metric space $L$ is complete with
respect to the probabilistic Pompeiu-Hausdorff metric $H$. The same is true for
the families of all closed bounded, respectively compact, nonempty subsets of
$L$. If $L$ is a complete random normed space in the sense of \v{S}erstnev,
then the family of all nonempty closed convex subsets of $L$ is also complete
with respect to $H$.
http://arXiv.org/abs/math/0507207
http://front.math.ucdavis.edu/math.PR/0507207
(alternate)
Author(s): Vincent Beffara (UMPA-ENSL) and Vladas Sidoravicius (IMPA)
Abstract: This is a survey article to be part of the Encyclopedia of Mathematical
Physics, to be published by Elsevier in the beginning of 2006.
http://arXiv.org/abs/math/0507220
http://front.math.ucdavis.edu/math.PR/0507220
(alternate)
Author(s): Marton Balazs and Firas Rassoul-Agha and Timo Seppalainen
Abstract: We study space-time fluctuations around a characteristic line for a
one-dimensional interacting system known as the random average process. The
state of this system is a real-valued function on the integers. New values of
the function are created by averaging previous values with random weights. The
fluctuations analyzed occur on the scale n^{1/4} where n is the ratio of
macroscopic and microscopic scales in the system. The limits of the
fluctuations are described by a family of Gaussian processes. In cases of known
product-form equilibria, this limit is a two-parameter process whose time
marginals are fractional Brownian motions with Hurst parameter 1/4. Along the
way we study the limits of quenched mean processes for a random walk in a
space-time random environment. These limits also happen at scale n^{1/4} and
are described by certain Gaussian processes that we identify. In particular,
when we look at a backward quenched mean process, the limit process is the
solution of a stochastic heat equation.
http://arXiv.org/abs/math/0507226
http://front.math.ucdavis.edu/math.PR/0507226
(alternate)
Author(s): Mikhail Kovtun and Igor Akushevich and Kenneth G. Manton and H. Dennis Tolley
Abstract: We present a new efficient algortithm for construction of linear latent
structure (LLS) models. This algorithm reduces a problem of estimation of model
parameters to a sequence of problems of linear algebra, which assures a low
computational complexity and ability to handle on desktop computers data that
involve up to thousands of variables.
http://arXiv.org/abs/math/0507021
http://front.math.ucdavis.edu/math.PR/0507021
(alternate)
Author(s): Setsuo Taniguchi
Abstract: The bijectivity of the mapping, which is represented as expectation, from a
family of Gaussian measures parametrized by linear combinations of Dirac
measures to the space of classical reflectionless potentials is shown. It is
also shown that the bijectivity extends to the space of generalized
reflectionless potentials, which was used by V. Marchenko to study the Cauchy
problem for the KdV equation. In the extension, the stochastic calculus based
on the Brownian sheet plays a key role.
http://arXiv.org/abs/math/0507229
http://front.math.ucdavis.edu/math.PR/0507229
(alternate)
Author(s): Guangyue Han and Brian Marcus
Abstract: We prove that under a mild positivity assumption the entropy rate of a hidden
Markov chain varies analytically as a function of the underlying Markov chain
parameters. We give examples to show how this can fail in some cases. And we
study two natural special classes of hidden Markov chains in more detail:
binary hidden Markov chains with an unambiguous symbol and binary Markov chains
corrupted by binary symmetric noise. Finally, we show that under the positivity
assumption the hidden Markov chain {\em itself} varies analytically, in a
strong sense, as a function of the underlying Markov chain parameters.
http://arXiv.org/abs/math/0507235
http://front.math.ucdavis.edu/math.PR/0507235
(alternate)
Author(s): F. Klebaner and R. Liptser
Abstract: We give an explicit formula for the most likely path to extinction for the
Galton-Watson processes with large initial population. We establish this result
with the help of the large deviation principle (LDP) which also recovers the
asymptotics of extinction probability.
Due to the nonnegativity of the Galton-Watson processes, the proof of LDP
verification at the point of extinction uses a nonstandard argument of
independent interest.
http://arXiv.org/abs/math/0507257
http://front.math.ucdavis.edu/math.PR/0507257
(alternate)
Author(s): F. Klebaner and R. Liptser
Abstract: We clarify the boundary effect in Cramer's theorem on the Large Deviations
Principle (LDP) for normed sums of non-negative i.i.d. random variables $
S_n=\frac{1}{n}\sum_{i=1}^n\xi_i $. We show that the LDP holds true with the
rate function possibly infinite at the boundary point $x=0$.
We also consider a continuous time version of Cramer's theorem with
nonnegative summands $ S_t=\frac{1}{t}\sum_{i:\tau_i\le t}\xi_i, t \to\infty, $
where $(\tau_i,\xi_i)_{i\ge 1}$ is a sequence of random variables such that
$tS_t$ is a random process with independent increments.
http://arXiv.org/abs/math/0507258
http://front.math.ucdavis.edu/math.PR/0507258
(alternate)
Author(s): Gopal K. Basak and Zhan-Qian Lu
Abstract: Switching ARMA models greatly enhance the standard linear models to the
extent that different ARMA model is allowed in a different regime, and the
regime switching is typically assumed a Markov chain on the finite states of
potential regimes. Although statistical issues have been the subject of many
recent papers, there is few systematic study of the probabilistic aspects of
this new class of nonlinear models. This paper discusses some basic issues
concerning this class of models including strict stationarity, influence of
initial conditions, and second-order property by studying SVAR models. A number
of examples are given to illustrate the theory and the variety of applications.
Extensions to other models such as mean-shifting, and inhomogeneous transition
probabilities are discussed.
http://arXiv.org/abs/math/0507267
http://front.math.ucdavis.edu/math.ST/0507267
(alternate)
Author(s): Eric van Fossen Conrad and Philippe Flajolet
Abstract: Elliptic functions considered by Dixon in the nineteenth century and related
to Fermat's cubic, $x^3+y^3=1$, lead to a new set of continued fraction
expansions with sextic numerators and cubic denominators. The functions and the
fractions are pregnant with interesting combinatorics, including a special
P\'olya urn, a continuous-time branching process of the Yule type, as well as
permutations satisfying various constraints that involve either parity of
levels of elements or a repetitive pattern of order three. The combinatorial
models are related to but different from models of elliptic functions earlier
introduced by Viennot, Flajolet, Dumont, and Fran{\c{c}}on.
http://arXiv.org/abs/math/0507268
http://front.math.ucdavis.edu/math.CO/0507268
(alternate)
Author(s): Joern Davidsen and Peter Grassberger and Maya Paczuski
Abstract: We extend the notion of waiting times for a point process to recurrent events
in space-time. Earthquake $B$ is a recurrence of a previous one, $A$, if no
intervening earthquake happens after $A$ and before $B$ in the spatial disc
centered on $A$ with radius $\bar{AB}$. The cascade of recurrent events, where
each later recurrence to an event is closer in space than all previous ones,
forms a sequence of records. Representing each record by a directed link
between nodes defines a network of earthquakes. For Southern California, this
network exhibits robust scaling laws. The rupture length emerges as a
fundamental scale for distance between recurrent events. Also, the distribution
of relative separations for the next record in space (time) $\sim
r^{-\delta_r}$ ($\sim t^{-\delta_t}$), with $\delta_r \approx \delta_t \approx
0.6$. While the in-degree distribution agrees with a random network, the
out-degree distribution shows large deviations from Poisson statistics.
Comparison with randomized data and a theory of records for independent events
occurring on a fractal shows that these statistics capture non-trivial features
of the complex spatiotemporal organization of seismicity.
http://arXiv.org/abs/physics/0507082
http://front.math.ucdavis.edu/physics/0507082
(alternate)
Author(s): Mikhail Kovtun and Igor Akushevich and Kenneth G. Manton and H. Dennis Tolley
Abstract: A new method for analyzing high-dimensional categorical data, Linear Latent
Structure (LLS) analysis, is presented. LLS models belong to the family of
latent structure models, which are mixture distribution models constrained to
satisfy the local independence assumption. LLS analysis explicitly considers a
family of mixed distributions as a linear space and LLS models are obtained by
imposing linear constraints on the mixing distribution. LLS models are
identifiable under modest conditions and are consistently estimable. A
remarkable feature of LLS analysis is the existence of a high-performance
numerical algorithm, which reduces parameter estimation to a sequence of linear
algebra problems. Preliminary simulation experiments with a prototype of the
algorithm demonstrated a good quality of restoration of model parameters.
http://arXiv.org/abs/math/0507025
http://front.math.ucdavis.edu/math.PR/0507025
(alternate)
Author(s): Julien Dubedat
Abstract: Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a
single continuous random conformally invariant interface in a simply connected
planar domain; the admissible probability distributions are parameterized by a
single positive parameter $\kappa$. As shown in \cite{D6}, the coexistence of
$n$ interfaces in such a domain implies algebraic ("commutation") conditions.
In the most interesting situations, the admissible laws on systems of $n$
interfaces are parameterized by $\kappa$ and the solution of particular (finite
rank) holonomic systems. The study of solutions of differential systems, in
particular their global behaviour, often involves the use of integral
representations. In the present article, we provide Euler integral
representations for solutions of holonomic systems arising from SLE
commutation. Applications to critical percolation (general crossing formulae),
loop-erased random walks (direct derivation of Fomin's formulae in the scaling
limit), and uniform spanning trees are discussed. The connection with conformal
restriction and Poissonized non-intersection for chordal SLEs is also studied.
http://arXiv.org/abs/math/0507276
http://front.math.ucdavis.edu/math.PR/0507276
(alternate)
Author(s): Wojciech Matysiak and Pawe{\l} J. Szab{\l}owski
Abstract: We examine problem of existence of stationary random fields with linear
regressions and quadratic conditional variances, introduced by Bryc in
"Stationary random fields with linear regressions" (Annals of Probability 29,
No. 1, 504-519). Distributions of the fields are identified and almost complete
description of the possible sets of parameters defining the first two
conditional moments is given. This note almost solves Bryc's problem concerning
fields undetermined by moments - the only remaining set of parameters for which
the existence of Bryc's fields is unclear has Lebesgue measure zero.
http://arXiv.org/abs/math/0507296
http://front.math.ucdavis.edu/math.PR/0507296
(alternate)
Author(s): Pawe{\l} J. Szab{\l}owski
Abstract: We define two Markov processes. The finite dimensional distributions of the
first one (say $\mathbf{X=}(X_{t})_{t\geq0})$ depend on one parameter
$q\in(-1,1>$ and of the second one (say $\mathbf{Y=}(Y_{t})_{t\in\mathbb{R}})$
on two parameters $(q,\alpha) \in(-1,1>\times(0,\infty).$ The first one
resembles Wiener process in the sense that for $q=1$ it is Wiener process but
also that for $q<1$ and $\forall n\geq1$ $t^{n/2}H_{n}(X_{t}/\sqrt{t}|q) ,$
where $(H_{n})_{n\geq0}$ are so called $q-$Hermite polynomials, are
martingales. It does not have however independent increments. The second one
resemble Orstein-Ulehnbeck processes. For $q=1$ it is a classical OU process.
For $q<1$ it is stationary with correlation function equal to $\exp
(-\alpha|t-s|).$When defining these processes and proving their existence we
use properties of discrete time Bryc processes and solve the problem of their
existence for $q>1.$ On the way we deny Wesolowski's martingale
characterization of Wiener process.
http://arXiv.org/abs/math/0507303
http://front.math.ucdavis.edu/math.PR/0507303
(alternate)
Author(s): Ben Morris
Abstract: The Thorp shuffle is defined as follows. Cut the deck into two equal piles.
Drop the first card from the left pile or the right pile according to the
outcome of a fair coin flip; then drop from the other pile. Continue this way
until both piles are empty. We show that the mixing time for the Thorp shuffle
with $2^d$ cards is polynomial in $d$.
http://arXiv.org/abs/math/0507307
http://front.math.ucdavis.edu/math.PR/0507307
(alternate)
Author(s): Christopher Hoffman and Alexander E. Holroyd and Yuval Peres
Abstract: Let \Xi be a discrete set in R^d. Call the elements of \Xi centers. The
well-known Voronoi tessellation partitions R^d into polyhedral regions (of
varying volumes) by allocating each site of R^d to the closest center. Here we
study allocations of R^d to \Xi in which each center attempts to claim a region
of equal volume \alpha.
We focus on the case where \Xi arises from a Poisson process of unit
intensity. It was proved in math.PR/0505668 that there is a unique allocation
which is stable in the sense of the Gale-Shapley marriage problem. We study the
distance X from a typical site to its allocated center in the stable
allocation.
The model exhibits a phase transition in the appetite \alpha. In the critical
case \alpha=1 we prove a power law upper bound on X in dimension d=1. It is an
open problem to prove any upper bound in d\geq 2. (Power law lower bounds were
proved in math.PR/0505668 for all d). In the non-critical cases \alpha<1 and
\alpha>1 we prove exponential upper bounds on X.
http://arXiv.org/abs/math/0507324
http://front.math.ucdavis.edu/math.PR/0507324
(alternate)
Author(s): Wojciech Matysiak and Pawe{\l} J. Szab{\l}owski
Abstract: The aim of the paper is to analyze square integrable random sequences
$\mathbf{X}=(X_{k})_{k\in\mathbb{Z}}$ satisfying condition \[
\wwo{X_k}{...,X_{k-2},X_{k-1},X_{k+1},X_{k+2},...}=\sum_{j=1}^n b_j
(X_{k-j}+X_{k+j}) \] with $b_{j}\in\mathbb{R}$ and
$n\in\nat\cup\left\{+\infty\right\}$. The question of existence of such
sequences for all $n\in\mathbb{N}$ including $n=+\infty$ is examined and some
conditions guaranteeing existence are provided. In order to give these
conditions we analyze general problem of existence of processes defined by
regression coefficients. The problem is closely related to one considered by
Kingman and Williams.
One of the results presented in the paper is that one sided regressions of
$\mathbf{X}$ are also linear: \[
\mathbb{E}(X_{k}|...,X_{k-2},X_{k-1})=\sum_{j=1}^{n}\beta_{j}X_{k -j}% \] for
some $\beta_{j}\in\mathbb{R}$ and with the same $n$ as before.
http://arXiv.org/abs/math/0507332
http://front.math.ucdavis.edu/math.PR/0507332
(alternate)
Author(s): Florent Benaych-Georges (DMA)
Abstract: We characterize asymptotic collective behaviour of rectangular random
matrices, the sizes of which tend to infinity at different rates: when embedded
in a space of larger square matrices, independent rectangular random matrices
are asymtotically free with amalgamation over a subalgebra. Therefore we can
define a "rectangular free convolution", linearized by cumulants and by an
analytic integral transform, called the "rectangular R-transform".
http://arXiv.org/abs/math/0507336
http://front.math.ucdavis.edu/math.PR/0507336
(alternate)
Author(s): Michael Erlihson and Boris Granovsly
Abstract: We find limit shapes for a family of multiplicative measures on the set of
partitions, induced by exponential generating functions with expansive
parameters, $a_k\sim k^{p-1}, k\to\infty, p>0$. The measures considered are
associated with reversible coagulation-fragmentation processes and certain
combinatorial structures. We prove the functional central limit theorem for the
fluctuations of a scaled random partition around its limit shape. We also
demonstrate that when the component size passes beyond the threshold value, the
independence of numbers of components transforms into their conditional
independence. Among other things, the paper discusses, in a general setting,
the interplay between limit shapes, threshold and gelation.
http://arXiv.org/abs/math/0507343
http://front.math.ucdavis.edu/math.PR/0507343
(alternate)
Author(s): Thomas M. Liggett
Abstract: A 1977 theorem of T. Harris states that an attractive spin system preserves
the class of associated probability measures. We study analogues of this result
for measures that satisfy various conditional positive c | |
