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Probability Abstracts 92
This document contains abstracts 4110-4254 from
Mar-1-2006 to Apr-28-2006.
They have been mailed on May 5, 2006.
Author(s): Jin Feng
Abstract: Large deviation for Markov processes can be studied by Hamilton--Jacobi
equation techniques. The method of proof involves three steps: First, we apply
a nonlinear transform to generators of the Markov processes, and verify that
limit of the transformed generators exists. Such limit induces a
Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness
(the comparison principle) holds for the limit equation. Finally, we verify an
exponential compact containment estimate. The large deviation principle then
follows from the above three verifications. This paper illustrates such a
method applied to a class of Hilbert-space-valued small diffusion processes.
The examples include stochastically perturbed Allen--Cahn, Cahn--Hilliard PDEs
and a one-dimensional quasilinear PDE with a viscosity term. We prove the
comparison principle using a variant of the Tataru method. We also discuss
different notions of viscosity solution in infinite dimensions in such context.
http://arXiv.org/abs/math/0602655
http://front.math.ucdavis.edu/math.PR/0602655
(alternate)
Author(s): Jean Bertoin (PMA) and Ronald A. Doney and Ross A. Maller (CMA)
Abstract: We wish to characterise when a L\'{e}vy process $X\_t$ crosses boundaries
like $t^\kappa$, $\kappa>0$, in a one or two-sided sense, for small times $t$;
thus, we enquire when $\limsup\_{t\downarrow 0}|X\_t|/t^{\kappa}$,
$\limsup\_{t\downarrow 0}X\_t/t^{\kappa}$ and/or $\liminf\_{t\downarrow
0}X\_t/t^{\kappa}$ are almost surely (a.s.) finite or infinite. Necessary and
sufficient conditions are given for these possibilities for all values of
$\kappa>0$. Often (for many values of $\kappa$), when the limsups are finite
a.s., they are in fact zero, as we show, but the limsups may in some
circumstances take finite, nonzero, values, a.s. In general, the process
crosses one or two-sided boundaries in quite different ways, but surprisingly
this is not so for the case $\kappa=1/2$. An integral test is given to
distinguish the possibilities in that case. Some results relating to other
norming sequences for $X$, and when $X$ is centered at a nonstochastic
function, are also given.
http://arXiv.org/abs/math/0603274
http://front.math.ucdavis.edu/math.PR/0603274
(alternate)
Author(s): Dmitry B. Rokhlin
Abstract: We obtain a constructive criterion for robust no-arbitrage in discrete-time
market models with transaction costs. This criterion is expressed in terms of
the supports of the regular conditional upper distributions of the solvency
cones. We also consider the model with a bank account. A method for
construction of arbitrage strategies is proposed.
http://arXiv.org/abs/math/0603284
http://front.math.ucdavis.edu/math.PR/0603284
(alternate)
Author(s): Rainer Siegmund-Schultze and Wolfgang Wagner
Abstract: A two-site spatial coagulation model is considered. Particles of masses m and
n at the same site form a new particle of mass m+n at rate mn. Independently,
particles jump to the other site at a constant rate. The limit (for increasing
particle numbers) of this model is expected to be non-deterministic after the
gelation time, namely, one or two giant particles randomly jump between the two
sites. Moreover, a new effect of induced gelation is observed - the gelation
happening at the site with the larger initial number of monomers immediately
induces gelation at the other site. Induced gelation is shown to be of
logarithmic order. The limiting behaviour of the model is derived rigorously up
to the gelation time, while the expected post-gelation behaviour is illustrated
by a numerical simulation.
http://arXiv.org/abs/math/0603300
http://front.math.ucdavis.edu/math.PR/0603300
(alternate)
Author(s): Marton Balazs and Eric Cator and Timo Seppalainen
Abstract: We study the last-passage growth model on the planar integer lattice with
exponential weights. With boundary conditions that represent the equilibrium
exclusion process as seen from a particle right after its jump we prove that
the variance of the last-passage time in a characteristic direction is of order
t^{2/3}. With more general boundary conditions that include the rarefaction fan
case we show that the last-passage time fluctuations are still of order
t^{1/3}, and also that the transversal fluctuations of the maximal path have
order t^{2/3}. We adapt and then build on a recent study of Hammersley's
process by Cator and Groeneboom, and also utilize the competition interface
introduced by Ferrari, Martin and Pimentel. The arguments are entirely
probabilistic, and no use is made of the combinatorics of Young tableaux or
methods of asymptotic analysis.
http://arXiv.org/abs/math/0603306
http://front.math.ucdavis.edu/math.PR/0603306
(alternate)
Author(s): E. Ben-Naim and P.L. Krapivsky
Abstract: We study how weak disorder affects the growth of the Fibonacci series. We
introduce a family of stochastic sequences that grow by the normal Fibonacci
recursion with probability 1-epsilon, but follow a different recursion rule
with a small probability epsilon. We focus on the weak disorder limit and
obtain the Lyapunov exponent, that characterizes the typical growth of the
sequence elements, using perturbation theory. The limiting distribution for the
ratio of consecutive sequence elements is obtained as well. A number of
variations to the basic Fibonacci recursion including shift, doubling, and
copying are considered.
http://arXiv.org/abs/cond-mat/0603117
http://front.math.ucdavis.edu/cond-mat/0603117
(alternate)
Author(s): Percy Deift
Abstract: All physical systems in equilibrium obey the laws of thermodynamics. In other
words, whatever the precise nature of the interaction between the atoms and
molecules at the microscopic level, at the macroscopic level, physical systems
exhibit universal behavior in the sense that they are all governed by the same
laws and formulae of thermodynamics. In this paper we describe some recent
history of universality ideas in physics starting with Wigner's model for the
scattering of neutrons off large nuclei and show how these ideas have led
mathematicians to investigate universal behavior for a variety of mathematical
systems. This is true not only for systems which have a physical origin, but
also for systems which arise in a purely mathematical context such as the
Riemann hypothesis, and a version of the card game solitaire called patience
sorting.
http://arXiv.org/abs/math-ph/0603038
http://front.math.ucdavis.edu/math-ph/0603038
(alternate)
Author(s): Andrei Agrachev (SISSA-Isas) and Sergei Kuksin (Mathematics Department of Heriot-Watt University), Andrey Sarychev (DMD), Armen Shirikyan (LM-Orsay)
Abstract: The paper is devoted to studying the image of probability measures on a
Hilbert space under finite-dimensional analytic maps. We establish sufficient
conditions under which the image of a measure has a density with respect to the
Lebesgue measure and continuously depends on the map. The results obtained are
applied to the 2D Navier--Stokes equations perturbed by various random forces
of low dimension.
http://arXiv.org/abs/math/0603295
http://front.math.ucdavis.edu/math.AP/0603295
(alternate)
Author(s): Maria A. Avino-Diaz and Gabriela Bulancea and Oscar Moreno
Abstract: In this paper we introduce the idea of probability in the definition of a
Sequential Dynamical System (SDS), thus obtaining a new concept, that of
Probabilistic Sequential System (PSS). Due to its particular dynamic, the
Probabilistic Boolean Network (PBN) model has been applied to genetic
regulatory networks. The model we introduce combines the sequential aspect of
the SDSs and the dynamic of the PBNs. The notion of simulation of a PSS is
introduced using the concept of morphism of PSSs. We prove that the PSSs with
the PSS-morphisms form a category PSS. Several examples of morphisms,
subsystems and simulations are given.
http://arXiv.org/abs/math/0603289
http://front.math.ucdavis.edu/math.DS/0603289
(alternate)
Author(s): Maria A. Avino-Diaz
Abstract: In this paper we study finite dynamical systems with $n$ functions acting on
the same set $X$, and probabilities assigned to these functions, that it is
called Probabilistic Regulatory Gene Networks (PRN. his concept is the same or
a natural generalization of the concept Probabilistic Boolean Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang, particularly the model
PBN has been using to describe genetic networks and has therapeutic
applications. In PRNs the most important question is to describe the steady
states of the systems, so in this paper we pay attention to the idea of
transforming a network to another without lost all the properties, in
particular the probability distribution. Following this objective we develop
the concepts of homomorphism and $\epsilon$-homomorphism of probabilistic
regulatory networks, since these concepts bring the properties from one
networks to another. Projections are special homomorphisms, and they always
induce invariant subnetworks that contain all cycles and steady states in the
network.
http://arXiv.org/abs/math/0603291
http://front.math.ucdavis.edu/math.DS/0603291
(alternate)
Author(s): Maria A. Avino-Diaz
Abstract: In this paper we study homomorphisms of Probabilistic Regulatory Gene
Networks(PRN) introduced in arXiv:math.DS/0603289 v1 13 Mar 2006. The model PRN
is a natural generalization of the Probabilistic Boolean Networks (PBN),
introduced by I. Shmulevich, E. Dougherty, and W. Zhang in 2001, that has been
using to describe genetic networks and has therapeutic applications. In this
paper, our main objectives are to apply the concept of homomorphism and
$\epsilon$-homomorphism of probabilistic regulatory networks to the dynamic of
the networks. The meaning of $\epsilon$ is that these homomorphic networks have
similar distributions and the distance between the distributions is upper
bounded by $\epsilon$. Additionally, we prove that the class of PRN together
with the homomorphisms form a category with products and coproducts.
Projections are special homomorphisms, and they always induce invariant
subnetworks that contain all the cycles and steady states in the network. Here,
it is proved that the $\epsilon$-homomorphism for $0<\epsilon<1$ produce
simultaneous Markov Chains in both networks, that permit to introduce the
concept of $\epsilon$-isomorphism of Markov Chains, and similar networks.
http://arXiv.org/abs/math/0603302
http://front.math.ucdavis.edu/math.DS/0603302
(alternate)
Author(s): Jaime A. Londo\~no
Abstract: We propose a new approach to utilities that is consistent with
state-dependent utilities. In our model utilities reflect the level of
consumption satisfaction of flows of cash in future times as they are valued
when the economic agents are making their consumption and investment decisions.
The theoretical framework used for the model is one proposed by the author in
Dynamic State Tameness {arXiv:math.PR/0509139}. The proposed framework is a
generalization of the theory of Brownian flows and can be applied to those
processes that are the solutions of classical It^o stochastic differential
equations, even when the volatilities and drifts are just locally
$\delta$-Holder continuous for some $\delta>0$. We develop the martingale
methodology for the solution of the problem of optimal consumption and
investment. Complete solutions of the optimal consumption and portfolio problem
are obtained in a very general setting which includes several functional forms
for utilities in the current literature, and consider general restrictions on
minimal wealths. As a secondary result we obtain a suitable representation for
straightforward numerical computations of the optimal consumption and
investment strategies.
http://arXiv.org/abs/math/0603316
http://front.math.ucdavis.edu/math.PR/0603316
(alternate)
Author(s): Martin Dyer and Leslie Ann Goldberg and Mark Jerrum
Abstract: We address the problem of sampling colorings of a graph $G$ by Markov chain
simulation. For most of the article we restrict attention to proper
$q$-colorings of a path on $n$ vertices (in statistical physics terms, the
one-dimensional $q$-state Potts model at zero temperature), though in later
sections we widen our scope to general ``$H$-colorings'' of arbitrary graphs
$G$. Existing theoretical analyses of the mixing time of such simulations
relate mainly to a dynamics in which a random vertex is selected for updating
at each step. However, experimental work is often carried out using systematic
strategies that cycle through coordinates in a deterministic manner, a dynamics
sometimes known as systematic scan. The mixing time of systematic scan seems
more difficult to analyze than that of random updates, and little is currently
known. In this article we go some way toward correcting this imbalance. By
adapting a variety of techniques, we derive upper and lower bounds (often
tight) on the mixing time of systematic scan. An unusual feature of systematic
scan as far as the analysis is concerned is that it fails to be time
reversible.
http://arXiv.org/abs/math/0603323
http://front.math.ucdavis.edu/math.PR/0603323
(alternate)
Author(s): Cedric Boutillier
Abstract: In this paper, we introduce a family of observables for the dimer model on a
bi-periodic bipartite planar graph, called pattern density fields. We study the
scaling limit of these objects for liquid and gaseous Gibbs measures of the
dimer model, and prove that they converge to a linear combination of a
derivative of the Gaussian massless free field and an independent white noise.
http://arXiv.org/abs/math/0603324
http://front.math.ucdavis.edu/math.PR/0603324
(alternate)
Author(s): D. R. McDonald and J. Reynier
Abstract: RED (Random Early Detection) has been suggested when multiple TCP sessions
are multiplexed through a bottleneck buffer. The idea is to detect congestion
before the buffer overflows by dropping or marking packets with a probability
that increases with the queue length. The objectives are reduced packet loss,
higher throughput, reduced delay and reduced delay variation achieved through
an equitable distribution of packet loss and reduced synchronization. Baccelli,
McDonald and Reynier [Performance Evaluation 11 (2002) 77--97] have proposed a
fluid model for multiple TCP connections in the congestion avoidance regime
multiplexed through a bottleneck buffer implementing RED. The window sizes of
each TCP session evolve like independent dynamical systems coupled by the queue
length at the buffer. The key idea in [Performance Evaluation 11 (2002) 77--97]
is to consider the histogram of window sizes as a random measure coupled with
the queue. Here we prove the conjecture made in [Performance Evaluation 11
(2002) 77--97] that, as the number of connections tends to infinity, this
system converges to a deterministic mean-field limit comprising the window size
density coupled with a deterministic queue.
http://arXiv.org/abs/math/0603325
http://front.math.ucdavis.edu/math.PR/0603325
(alternate)
Author(s): Sean P. Meyn
Abstract: Consider the normalized partial sums of a real-valued function $F$ of a
Markov chain, \[\phi_n:=n^{-1}\sum_{k=0}^{n-1}F(\Phi(k)),\qquad n\ge1.\] The
chain $\{\Phi(k):k\ge0\}$ takes values in a general state space $\mathsf {X}$,
with transition kernel $P$, and it is assumed that the Lyapunov drift condition
holds: $PV\le V-W+b\mathbb{I}_C$ where $V:\mathsf {X}\to(0,\infty)$, $W:\mathsf
{X}\to[1,\infty)$, the set $C$ is small and $W$ dominates $F$. Under these
assumptions, the following conclusions are obtained: 1. It is known that this
drift condition is equivalent to the existence of a unique invariant
distribution $\pi$ satisfying $\pi(W)<\infty$, and the law of large numbers
holds for any function $F$ dominated by $W$:
\[\phi_n\to\phi:=\pi(F),\qquad{a.s.}, n\to\infty.\] 2. The lower error
probability defined by $\mathsf {P}\{\phi_n\le c\}$, for $c<\phi$, $n\ge1$,
satisfies a large deviation limit theorem when the function $F$ satisfies a
monotonicity condition. Under additional minor conditions an exact large
deviations expansion is obtained. 3. If $W$ is near-monotone, then
control-variates are constructed based on the Lyapunov function $V$, providing
a pair of estimators that together satisfy nontrivial large asymptotics for the
lower and upper error probabilities. In an application to simulation of queues
it is shown that exact large deviation asymptotics are possible even when the
estimator does not satisfy a central limit theorem.
http://arXiv.org/abs/math/0603328
http://front.math.ucdavis.edu/math.PR/0603328
(alternate)
Author(s): Teddy Seidenfeld and Mark J. Schervish and Joseph B. Kadane
Abstract: Correction to Annals of Probability 29 (2001) 1612--1624
[doi:10.1214/aop/1015345764].
http://arXiv.org/abs/math/0603012
http://front.math.ucdavis.edu/math.PR/0603012
(alternate)
Author(s): Li-X. Zhang and Feifang Hu and Siu Hung Cheung
Abstract: The Generalized P\'{o}lya Urn (GPU) is a popular urn model which is widely
used in many disciplines. In particular, it is extensively used in treatment
allocation schemes in clinical trials. In this paper, we propose a sequential
estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum
of applications. Because the proposed urn model depends on sequential
estimations of unknown parameters, the derivation of asymptotic properties is
mathematically intricate and the corresponding results are unavailable in the
literature. We overcome these hurdles and establish the strong consistency and
asymptotic normality for both the patient allocation and the estimators of
unknown parameters, under some widely satisfied conditions. These properties
are important for statistical inferences and they are also useful for the
understanding of the urn limiting process. A superior feature of our proposed
model is its capability to yield limiting treatment proportions according to
any desired allocation target. The applicability of our model is illustrated
with a number of examples.
http://arXiv.org/abs/math/0603329
http://front.math.ucdavis.edu/math.PR/0603329
(alternate)
Author(s): Stan Zachary and Serguei Foss
Abstract: We study the distribution of the maximum $M$ of a random walk whose
increments have a distribution with negative mean and belonging, for some
$\gamma\ge0$, to the class $\mathcal{S}_{\gamma}$ introduced by Chover, Ney,
and Weinger (1973). For $\gamma>0$, we give a probabilistic derivation of the
asymptotic tail distribution of $M$ and show that, as in the case $\gamma=0$,
extreme values of $M$ are in general attained through some single large
increment in the random walk.
http://arXiv.org/abs/math/0603330
http://front.math.ucdavis.edu/math.PR/0603330
(alternate)
Author(s): L. Belhadji and N. Lanchier
Abstract: Stochastic modeling of disease dynamics has had a long tradition. Among the
first epidemic models including a spatial structure in the form of local
interactions is the contact process. In this article we investigate two
extensions of the contact process describing the course of a single disease
within a spatially structured human population distributed in social clusters.
That is, each site of the $d$-dimensional integer lattice is occupied by a
cluster of individuals; each individual can be healthy or infected. The
evolution of the disease depends on three parameters, namely the outside
infection rate which models the interactions between the clusters, the within
infection rate which takes into account the repeated contacts between
individuals in the same cluster, and the size of each social cluster. For the
first model, we assume cluster recoveries, while individual recoveries are
assumed for the second one. The aim is to investigate the existence of
nontrivial stationary distributions for both processes depending on the value
of each of the three parameters. Our results show that the probability of an
epidemic strongly depends on the recovery mechanism.
http://arXiv.org/abs/math/0603331
http://front.math.ucdavis.edu/math.PR/0603331
(alternate)
Author(s): David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard Ycart (LMC - IMAG)
Abstract: For an $n\times n$ random image with independent pixels, black with
probability $p(n)$ and white with probability $1-p(n)$, the probability of
satisfying any given first-order sentence tends to 0 or 1, provided both
$p(n)n^{\frac{2}{k}}$ and $(1-p(n))n^{\frac{2}{k}}$ tend to 0 or $+\infty$, for
any integer $k$. The result is proved by computing the threshold function for
basic local sentences, and applying Gaifman's theorem.
http://arXiv.org/abs/math/0603333
http://front.math.ucdavis.edu/math.PR/0603333
(alternate)
Author(s): Deli Li and Andrew Rosalsky
Abstract: Let $\{X_{k,i};i\geq 1,k\geq 1\}$ be an array of i.i.d. random variables and
let $\{p_n;n\geq 1\}$ be a sequence of positive integers such that $n/p_n$ is
bounded away from 0 and $\infty$. For $W_n=\max_{1\leq i1/2)$, (ii) $\lim_{n\to
\infty}n^{1-\alpha}L_n=0$ a.s. $(1/2<\alpha \leq 1)$, (iii) $\lim_{n\to
\infty}\frac{W_n}{\sqrt{n\log n}}=2$ a.s. and (iv) $\lim_{n\to
\infty}(\frac{n}{\log n})^{1/2}L_n=2$ a.s. are shown to hold under optimal sets
of conditions. These results follow from some general theorems proved for
arrays of i.i.d. two-dimensional random vectors. The converses of the limit
laws (i) and (iii) are also established. The current work was inspired by
Jiang's study of the asymptotic behavior of the largest entries of sample
correlation matrices.
http://arXiv.org/abs/math/0603334
http://front.math.ucdavis.edu/math.PR/0603334
(alternate)
Author(s): N. Lanchier and C. Neuhauser
Abstract: Mutualists and pathogens, collectively called symbionts, are ubiquitous in
plant communities. While some symbionts are highly host-specific, others
associate with multiple hosts. The outcomes of multispecies host-symbiont
interactions with different degrees of specificity are difficult to predict at
this point due to a lack of a general conceptual framework. Complicating our
predictive power is the fact that plant populations are spatially explicit, and
we know from past research that explicit space can profoundly alter plant-plant
interactions. We introduce a spatially explicit, stochastic model to
investigate the role of explicit space and host-specificity in multispecies
host-symbiont interactions. We find that in our model, pathogens can
significantly alter the spatial structure of plant communities, promoting
coexistence, whereas mutualists appear to have only a limited effect. Effects
are more pronounced the more host-specific symbionts are.
http://arXiv.org/abs/math/0603335
http://front.math.ucdavis.edu/math.PR/0603335
(alternate)
Author(s): David Coupier (MAP5) and Agn\`{e}s Desolneux (MAP5) and Bernard Ycart (LMC - IMAG)
Abstract: Area openings and closings are morphological filters which efficiently
suppress impulse noise from an image, by removing small connected components of
level sets. The problem of an objective choice of threshold for the area
remains open. Here, a mathematical model for random images will be considered.
Under this model, a Poisson approximation for the probability of appearance of
any local pattern can be computed. In particular, the probability of observing
a component with size larger than $k$ in pure impulse noise has an explicit
form. This permits the definition of a statistical test on the significance of
connected components, thus providing an explicit formula for the area threshold
of the denoising filter, as a function of the impulse noise probability
parameter. Finally, using threshold decomposition, a denoising algorithm for
grey level images is proposed.
http://arXiv.org/abs/math/0603337
http://front.math.ucdavis.edu/math.PR/0603337
(alternate)
Author(s): Gerard Ben Arous and Jiri Cerny
Abstract: We give a general proof of aging for trap models using the arcsine law for
stable subordinators. This proof is based on abstract conditions on the
potential theory of the underlying graph and on the randomness of the trapping
landscape. We apply this proof to aging for trap models on large
two-dimensional tori and for trap dynamics of the Random Energy Model on a
broad range of time scales.
http://arXiv.org/abs/math/0603340
http://front.math.ucdavis.edu/math.PR/0603340
(alternate)
Author(s): Jir\^o Akahori
Abstract: This is a survey note of the author's observations on the discrete-time
analogues of It\^o formulas.
http://arXiv.org/abs/math/0603341
http://front.math.ucdavis.edu/math.PR/0603341
(alternate)
Author(s): Gerard Ben Arous and Jiri Cerny
Abstract: These notes cover one of the topics of the class given in the Les Houches
Summer School ``Mathematical statistical physics'' in July 2005. The lectures
tried to give a summary of the recent mathematical results about the long-time
behaviour of dynamics of (mean-field) spin-glasses and other disordered media.
We have chosen here to restrict the scope of these notes to the dynamics of
trap models only, but to cover this topic in somewhat more depth.
http://arXiv.org/abs/math/0603344
http://front.math.ucdavis.edu/math.PR/0603344
(alternate)
Author(s): S. Sethuraman
Abstract: Correction to Annals of Probability 28 (2000) 277--302
[doi:10.1214/aop/1019160120].
http://arXiv.org/abs/math/0603014
http://front.math.ucdavis.edu/math.PR/0603014
(alternate)
Author(s): Eric Cator and Piet Groeneboom
Abstract: We show that, for a stationary version of Hammersley's process, with Poisson
sources on the positive x-axis and Poisson sinks on the positive y-axis, the
variance of the length of a longest weakly North-East path $L(t,t)$ from
$(0,0)$ to $(t,t)$ is equal to $2\E(t-X(t))_+$, where $X(t)$ is the location of
a second class particle at time $t$. This implies that both $\E(t-X(t))_+$ and
the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the
relation between the flux and the path of a second class particle, continuing
the approach of Cator and Groeneboom (2005).
http://arXiv.org/abs/math/0603345
http://front.math.ucdavis.edu/math.PR/0603345
(alternate)
Author(s): Marcelo Sobottka
Abstract: In this paper we consider cellular automata $(\mathfrak{G},\Phi)$ with
algebraic local rules and such that $\mathfrak{G}$ is a topological Markov
chain which has a structure compatible to this local rule. We characterize such
cellular automata and study the convergence of the Ces\`aro mean distribution
of the iterates of any probability measure with complete connections and
summable decay.
http://arXiv.org/abs/math/0603326
http://front.math.ucdavis.edu/math.DS/0603326
(alternate)
Author(s): Yueyun Hu (LAGA) and Zhan Shi (PMA)
Abstract: We are interested in the random walk in random environment on an infinite
tree. Lyons and Pemantle [11] give a precise recurrence/transience criterion.
Our paper focuses on the almost sure asymptotic behaviours of a recurrent
random walk $(X\_n)$ in random environment on a regular tree, which is closely
related to Mandelbrot [13]'s multiplicative cascade. We prove, under some
general assumptions upon the distribution of the environment, the existence of
a new exponent $\nu\in (0, {1\over 2}]$ such that $\max\_{0\le i \le n} |X\_i|$
behaves asymptotically like $n^{\nu}$. The value of $\nu$ is explicitly
formulated in terms of the distribution of the environment.
http://arXiv.org/abs/math/0603363
http://front.math.ucdavis.edu/math.PR/0603363
(alternate)
Author(s): Cristian Coletti and Leandro P. R. Pimentel
Abstract: We study the asymptotics of beta-paths in the Hammersley process with sources
and sinks, in the rarefaction regime. We derive a strong law of large number
for those paths and we show that its fluctuation exponent is at most 2/3.
Examples of beta-paths are the space-time path of a second-class particle in
the Hammersley process and also the space-time path of the interface between
two PNG droplets.
http://arXiv.org/abs/math/0603382
http://front.math.ucdavis.edu/math.PR/0603382
(alternate)
Author(s): Jorge R. Busch and Pablo A. Ferrari and A. Georgina Flesia and Ricardo Fraiman and Sebastian Grynberg
Abstract: To distinguish between populations of trees, we consider the hypothesis test
proposed recently by Balding, Ferrari, Fraiman and Sued (BFFS--test). A direct
approach to calculate effectively the test statistic is quite difficult, since
it is based on a supremum defined over the space of all trees, which grows
exponentially fast. We show how to transform this problem into a max-flow over
a network which can be solved using a Ford Fulkerson algorithm in polynomial
time on the maximal number of vertices of the random tree. We also describe
conditions that imply the characterization of the measure by the marginal
distributions of each node of the random tree, which validate the use of the
BFFS--test for measure discrimination. The performance of the test is studied
via simulations on Galton-Watson processes.
http://arXiv.org/abs/math/0603378
http://front.math.ucdavis.edu/math.ST/0603378
(alternate)
Author(s): Patrick Cattiaux (CMAP and Modal'x) and Arnaud Guillin (CEREMADE)
Abstract: In this paper we derive non asymptotic deviation bounds for $$\P_\nu (|\frac
1t \int_0^t V(X_s) ds - \int V d\mu | \geq
R)$$ where $X$ is a $\mu$ stationary and ergodic Markov process and $V$ is
some $\mu$ integrable function. These bounds are obtained under various moments
assumptions for $V$, and various regularity assumptions for $\mu$. Regularity
means here that $\mu$ may satisfy various functional inequalities (F-Sobolev,
generalized Poincar\'e etc...).
http://arXiv.org/abs/math/0603021
http://front.math.ucdavis.edu/math.PR/0603021
(alternate)
Author(s): Francis Comets (PMA)
Abstract: In this paper, we consider directed polymers in random environment with long
range jumps in discrete space and time. We extend to this case some techniques,
results and classifications known in the usual short range case. However, some
properties are drastically different when the underlying random walk belongs to
the domain of attraction of an $\a$-stable law. For instance, we construct
natural examples of directed polymers in random environment which experience
weak disorder in low dimension.
http://arXiv.org/abs/math/0603390
http://front.math.ucdavis.edu/math.PR/0603390
(alternate)
Author(s): Alexander Roitershtein
Abstract: We consider transient random walks on a strip in a random environment. The
model was introduced by Bolthausen and Goldsheid in [4]. We derive a strong law
of large numbers for the random walks in a general ergodic setup and obtain an
annealed central limit theorem in the case of uniformly mixing environments. In
addition, we prove that the law of the ``environment viewed from the position
of the walker'' converges to a limiting distribution if the environment is an
i.i.d. sequence.
http://arXiv.org/abs/math/0603392
http://front.math.ucdavis.edu/math.PR/0603392
(alternate)
Author(s): Peter Eichelsbacher and Tomasz Schreiber
Abstract: Functionals of spatial point process often satisfy a weak spatial dependence
condition known as stabilization. In this paper we prove process level moderate
deviation principles (MDP) for such functionals, which are a level-3 result for
empirical point fields as well as a level-2 result for empirical point
measures. The level-3 rate function coincides with the so-called specific
information. We show that the general result can be applied to prove MDPs for
various particular functionals, including random sequential packing,
birth-growth models, germ-grain models and nearest neighbor graphs.
http://arXiv.org/abs/math/0603402
http://front.math.ucdavis.edu/math.PR/0603402
(alternate)
Author(s): Sergio De Carvalho Bezerra (IECN) and Samy Tindel (IECN) and Frederi Viens
Abstract: This paper provides information about the asymptotic behavior of a
one-dimensional Brownian polymer in random medium represented by a space-time
Gaussian field W assumed to be white noise in time and function-valued in
space. According to the behavior of the spatial covariance W, we give sharp
upper and lower bounds on the partition function's exponential rate (Lyapunov
exponent), and on the growth (wandering exponent) of the polymer itself when
the time parameter goes to infinity.
http://arXiv.org/abs/math/0603404
http://front.math.ucdavis.edu/math.PR/0603404
(alternate)
Author(s): F. Klebaner and R. Liptser
Abstract: The Large Deviation Principle is established for stochastic models defined by
past-dependent non linear recursions with small noise. In the Markov case we
use the result to obtain an explicit expression for the asymptotics of exit
time.
http://arXiv.org/abs/math/0603407
http://front.math.ucdavis.edu/math.PR/0603407
(alternate)
Author(s): Piotr Sniady
Abstract: We study the shape of the Young diagram \lambda associated via the
Robinson-Schensted-Knuth algorithm to a random permutation in S_n such that the
length of the longest decreasing subsequence is not bigger than a fixed number
d; in other words we study the restriction of the Plancherel measure to Young
diagrams with at most d rows. We prove that in the limit n\to\infty the rows of
\lambda behave like the eigenvalues of a certain random matrix (traceless
Gaussian Unitary Ensemble) with d rows and columns. In particular, the length
of the longest increasing subsequence of such a random permutation behaves
asymptotically like the largest eigenvalue of the corresponding random matrix.
http://arXiv.org/abs/math/0603401
http://front.math.ucdavis.edu/math.CO/0603401
(alternate)
Author(s): Daniela Bertacchi and Fabio Zucca
Abstract: We study weak and strong survival for branching random walks on multigraphs.
We prove that, for a large class of multigraphs, weak survival is related to a
geometrical parameter of the multigraph and that the existence of a pure weak
phase is equivalent to nonamenability. Finally we study weak and strong
critical behaviors of the branching random walk.
http://arXiv.org/abs/math/0603412
http://front.math.ucdavis.edu/math.PR/0603412
(alternate)
Author(s): Peter Eichelsbacher and Tomasz Schreiber and Joseph E. Yukich
Abstract: Functionals in geometric probability are often expressed as sums of bounded
functions exhibiting exponential stabilization.
Methods based on cumulant techniques and exponential modifications of measures
show that such functionals satisfy moderate deviation principles. This leads to
moderate deviation principles and laws of the iterated logarithm for random
packing models as well as for statistics associated with germ-grain models and
$k$ nearest neighbor graphs.
http://arXiv.org/abs/math/0603022
http://front.math.ucdavis.edu/math.PR/0603022
(alternate)
Author(s): Ilya Molchanov
Abstract: It is shown that max-stable random vectors in $[0,\infty)^d$ with unit
Fr\'echet marginals are in one to one correspondence with convex sets $K$ in
$[0,\infty)^d$ called max-zonoids. The max-zonoids can be characterised as sets
obtained as limits of Minkowski sums of simplices or, alternatively, as the
selection expectation of a random simplex whose distribution is controlled by
the spectral measure of the max-stable random vector. Furthermore, the
cumulative distribution function $\Prob{\xi\leq x}$ of a max-stable random
vector $\xi$ with unit Fr\'echet marginals is determined by the norm of the
inverse to $x$, where all possible norms are given by the support functions of
max-zonoids. As an application, geometrical interpretations of a number of
well-known concepts from the theory of multivariate extreme values and copulas
are provided. The convex geometry approach makes it possible to generalise a
number of known results and to introduce new operations with max-stable random
vectors.
http://arXiv.org/abs/math/0603423
http://front.math.ucdavis.edu/math.PR/0603423
(alternate)
Author(s): Philippe Briand (IRMAR) and Fulvia Confortola
Abstract: The aim of the present paper is to study the regularity properties of the
solution of a backward stochastic differential equation with a monotone
generator in infinite dimension. We show some applications to the nonlinear
Kolmogorov equation and to stochastic optimal control.
http://arXiv.org/abs/math/0603428
http://front.math.ucdavis.edu/math.PR/0603428
(alternate)
Author(s): Samuel Elogne and Dionisis Hristopulos
Abstract: This paper focuses on the estimation of model parameters (model inference)
for the class of Spartan Spatial Random Fields (SSRFs) introduced by
Hristopulos (2003). The approach used for model inference involves calculation
of sample constraints and fitting with respective ensemble constraints. The
fitting leads to optimal SSRF parameters obtained by minimizing a suitable
distance functional. We propose kernel-based estimators for calculating the
sample constraints from data distributed on irregular sampling grids. We
investigate the asymptotic properties of the estimators, and we establish a
criterion for the selection of the kernel bandwidth parameters. The performance
of the sample constraint estimators, as well as that of the SSRF inference
procedure is evaluated by means of numerical simulations for different models
of spatial dependence.
http://arXiv.org/abs/math/0603430
http://front.math.ucdavis.edu/math.ST/0603430
(alternate)
Author(s): Lorenzo Zambotti
Abstract: We consider stochastic differential equations in a Hilbert space, perturbed
by the gradient of a convex potential. We investigate the problem of
convergence of a sequence of such processes. We propose applications of this
method to reflecting O.U. processes in infinite dimension, to stochastic
partial differential equations with reflection of Cahn-Hilliard type and to
interface models.
http://arXiv.org/abs/math/0603474
http://front.math.ucdavis.edu/math.PR/0603474
(alternate)
Author(s): Feliks Przytycki and Juan Rivera-Letelier
Abstract: We study geometric and statistical properties of complex rational maps
satisfying the Topological Collet-Eckmann Condition. We show that every such a
rational map possesses a unique conformal probability measure of minimal
exponent, and that this measure is non-atomic, ergodic and that its Hausdorff
dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we
show that there is a unique invariant probability measure that is absolutely
continuous with respect to this conformal measure, and we show that this
measure is exponentially mixing (it has exponential decay of correlations) and
that it satisfies the Central Limit Theorem.
We also show that for a complex rational map f the existence of such an
invariant measure characterizes the Topological Collet-Eckmann Condition, and
that this measure is the unique equilibrium state with potential - HD(J(f)) ln
|f'|.
http://arXiv.org/abs/math/0603459
http://front.math.ucdavis.edu/math.DS/0603459
(alternate)
Author(s): Iosif Pinelis
Abstract: Let $\eta_1,\eta_2,...$ be independent (but not necessarily identically
distributed) zero-mean random variables (r.v.'s) such that $|\eta_i|\le1$
almost surely for all $i$, and let $Z$ stand for a standard normal r.v. Let
$a_1,a_2,...$ be any real numbers such that $a_1^2+a_2^2+...=1.$ It is shown
that then for all $x>0$ $$ \P(a_1\eta_1+a_2\eta_2+...\ge x) \le \P(Z\ge
x-\la/x), $$ where $\la := \ln\frac{2e^3}9=1.495...$. The proof relies on (i)
another probability inequality and (ii) a l'Hospital-type rule for
monotonicity, both developed elsewhere. Extensions to (super)martingales are
indicated.
http://arXiv.org/abs/math/0603030
http://front.math.ucdavis.edu/math.PR/0603030
(alternate)
Author(s): Faouzi Chaabane and Ahmed Kebaier
Abstract: We develop a general approach of the almost sure central limit theorem for
the quasi-continuous vectorial martingales and we release a quadratic extension
of this theorem while specifying speeds of convergence. As an application of
this result we study the problem of estimate the variance of a process with
stationary and idependent increments in statistics.
http://arXiv.org/abs/math/0603492
http://front.math.ucdavis.edu/math.PR/0603492
(alternate)
Author(s): Andrew R. Wade
Abstract: We give laws of large numbers (in the L^p sense) for the total length of the
k-nearest neighbours (directed) graph and the j-th nearest neighbour (directed)
graph in R^d, with power-weighted edges. We deduce a law of large numbers for
the standard nearest neighbour (undirected) graph. We give the limiting
constants, in the case of uniform random points in (0,1)^d, explicitly. Also,
we give explicit laws of large numbers for the total power-weighted length of
the Gabriel graph and two further graphs that are related to the standard
nearest-neighbour graph: the on-line nearest-neighbour graph and the minimal
directed spanning forest.
http://arXiv.org/abs/math/0603559
http://front.math.ucdavis.edu/math.PR/0603559
(alternate)
Author(s): Mathew D. Penrose and Andrew R. Wade
Abstract: In the on-line nearest-neighbour graph (ONG), each point after the first in a
sequence of points in R^d is joined by an edge to its nearest-neighbour amongst
those points that precede it in the sequence. We study the large-sample
asymptotic behaviour of the total power-weighted length of the ONG on uniform
random points in (0,1)^d. In particular, for d=1 and weight exponent
\alpha>1/2, the limiting distribution of the centred total weight is
characterized by a distributional fixed-point equation. As an ancillary result,
we give exact expressions for the expectation and variance of the standard
nearest-neighbour (directed) graph on uniform random points in the unit
interval.
http://arXiv.org/abs/math/0603561
http://front.math.ucdavis.edu/math.PR/0603561
(alternate)
Author(s): Sabir Umarov and Stanly Steinberg and Constantino Tsallis
Abstract: As well known, the standard central limit theorem plays a fundamental role in
Boltzmann-Gibbs (BG) statistical mechanics. This important physical theory has
been generalized by one of us (CT) in 1988 by using the entropy $S_q =
\frac{1-\sum_i p_i^q}{q-1}$ (with $q \in \cal{R}$) instead of its particular
case $S_1=S_{BG}= -\sum_i p_i \ln p_i$. The theory which emerges is usually
referred to as {\it nonextensive statistical mechanics} and recovers the
standard theory for $q=1$. During the last two decades, this $q$-generalized
statistical mechanics has been successfully applied to a considerable amount of
physically interesting complex phenomena. Conjectures and numerical indications
available in the literature were since a few years suggesting the possibility
of $q$-generalizations of the standard central limit theorem by allowing the
random variables that are being summed to be correlated in some special manner,
the case $q=1$ corresponding to standard probabilistic independence. This is
precisely what we prove in the present paper for some range of $q$ which
extends from below to above $q=1$. The attractor, in the usual sense of a
central limit theorem, is given by a distribution of the form $p(x) \propto
[1-(1-q) \beta x^2]^{1/(1-q)}$ with $\beta>0$. These distributions, sometimes
referred to as $q$-Gaussians, are known to make, under appropriate constraints,
extremal the functional $S_q$. Their $q=1$ and $q=2$ particular cases recover
respectively Gaussian and Cauchy distributions.
http://arXiv.org/abs/cond-mat/0603593
http://front.math.ucdavis.edu/cond-mat/0603593
(alternate)
Author(s): Xian-Yuan Wu and Yu Zhang
Abstract: We consider the supercritical oriented percolation model. Let ${\fK}$ be all
the percolation points. For each $u\in {\fK}$, we write $\gamma_u$ as its
right-most path. Let $G=\cup_u \gamma_u$. In this paper, we show that
$G$ is a single tree with only one topological end. We also present a
relationship between ${\fK}$ and $G$ and construct a bijection between ${\fK}$
and $\Z$ using the preorder traversal algorithm. Through applications of this
fundamental graph property, we show the uniqueness of an infinite oriented
cluster by ignoring finite vertices.
http://arXiv.org/abs/math/0603580
http://front.math.ucdavis.edu/math.PR/0603580
(alternate)
Author(s): A.Yu.Khrennikov and S.V.Kozyrev
Abstract: Gaussian random field on general ultrametric space is introduced as a
solution of pseudodifferential stochastic equation. Covariation of the
introduced random field is computed with the help of wavelet analysis on
ultrametric spaces.
Notion of ultrametric Markovianity, which describes independence of
contributions to random field from different ultrametric balls is introduced.
We show that the random field under investigation satisfies this property.
http://arXiv.org/abs/math/0603584
http://front.math.ucdavis.edu/math.PR/0603584
(alternate)
Author(s): Michael Blank
Abstract: We study raw coding of trajectories of a chaotic dynamical system by
sequences of elements from a finite alphabet and show that there is a
fundamental constraint on differences between codes corresponding to different
trajectories of the dynamical system.
http://arXiv.org/abs/math/0603575
http://front.math.ucdavis.edu/math.DS/0603575
(alternate)
Author(s): Maria Emilia Caballero and Lo\"{i}c Chaumont (PMA)
Abstract: By killing a stable L\'{e}vy process when it leaves the positive half-line,
or by conditioning it to stay positive, or by conditioning it to hit 0
continuously, we obtain three different positive self-similar Markov processes
which illustrate the three classes described by Lamperti \cite{La}. For each of
these processes, we compute explicitly the infinitesimal generator from which
we deduce the characteristics of the underlying L\'{e}vy process in the
Lamperti representation. The proof of this result bears on the behaviour at
time 0 of stable L\'{e}vy processes before their first passage time across
level 0 which we describe here. As an application, we give the law of the
minimum before an independent exponential time of a certain class of L\'{e}vy
processes. It provides the explicit form of the spacial Wiener-Hopf factor at a
particular point and the value of the ruin probability for this class of
L\'{e}vy processes.
http://arXiv.org/abs/math/0603613
http://front.math.ucdavis.edu/math.PR/0603613
(alternate)
Author(s): B. Siudeja
Abstract: Let $\alpha\in(0,2)$ and $X_t$ be a symmetric $\alpha$-stable process. We
define the scattering length $\Gamma(v)$ of the positive potential $v$ and
prove several of its basic properties. We use the scattering length to
findestimates for the first eigenvalue of the Schr\"odinger operator of the
``Neumann'' fractional Laplacian in a cube with potential $v$.
http://arXiv.org/abs/math/0603627
http://front.math.ucdavis.edu/math.PR/0603627
(alternate)
Author(s): Enzo Marinari and Guilhem Semerjian
Abstract: We apply in this article (non rigorous) statistical mechanics methods to the
problem of counting long circuits in graphs. The outcomes of this approach have
two complementary flavours. On the algorithmic side, we propose an approximate
counting procedure, valid in principle for a large class of graphs. On a more
theoretical side, we study the typical number of long circuits in random graph
ensembles, reproducing rigorously known results and stating new conjectures.
http://arXiv.org/abs/cond-mat/0603657
http://front.math.ucdavis.edu/cond-mat/0603657
(alternate)
Author(s): Q. S. Song and G. Yin
Abstract: This work establishes sufficient conditions for existence of saddle points in
discrete Markov games. The result reveals the relation between dynamic games
and static games using dynamic programming equations. This result enables us to
prove existence of saddle points of non-separable stochastic differential games
of regime-switching diffusions under appropriate conditions.
http://arXiv.org/abs/math/0603600
http://front.math.ucdavis.edu/math.OC/0603600
(alternate)
Author(s): Michael Voit
Abstract: Bessel-type convolution algebras of bounded Borel measures on the matrix
cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C,
\mathbb H$ were introduced recently by R\"osler. These convolutions depend on
some continuous parameter, generate commutative hypergroup structures and have
Bessel functions of matrix argument as characters.
Here, we first study the rich algebraic structure of these hypergroups. In
particular, the subhypergroups and automorphisms are classified, and we show
that each quotient by a subhypergroup carries a hypergroup structure of the
same type.
The algebraic properties are partially related to properties of random walks
on matrix Bessel hypergroups. In particular, known properties of Wishart
distributions, which form Gaussian convolution semigroups on these hypergroups,
are put into a new light. Moreover, limit theorems for random walks on these
hypergroups are presented. In particular, we obtain strong laws of large
numbers and a central limit theorem with Wishart distributions as limits.
http://arXiv.org/abs/math/0603017
http://front.math.ucdavis.edu/math.CA/0603017
(alternate)
Author(s): Jorge A. Leon and Jaime San Martin
Abstract: In this paper we use the chaos decomposition approach to establish the
existence of a unique continuous solution to linear fractional differential
equations of the Skorohod type. Here the coefficients are deterministic, the
inital condition is anticipating and the underlying fractional Brownian motion
has Hurst parameter less than 1/2. We provide an explicit expression for the
chaos decomposition of the solution in order to show our results.
http://arXiv.org/abs/math/0603636
http://front.math.ucdavis.edu/math.PR/0603636
(alternate)
Author(s): Erkan Nane
Abstract: Let $\tau_{D}(Z) $ be the first exit time of iterated Brownian motion from a
domain $D \subset \RR{R}^{n}$ started at $z\in D$ and let $P_{z}[\tau_{D}(Z)
>t]$ be its distribution. In this paper we establish the exact asymptotics of
$P_{z}[\tau_{D}(Z) >t]$ over bounded domains as an improvement of the results
in \cite{deblassie, nane2}, for $z\in D$ \begin{eqnarray}
\lim_{t\to\infty} t^{-1/2}\exp({3/2}\pi^{2/3}\lambda_{D}^{2/3}t^{1/3})
P_{z}[\tau_{D}(Z)>t]= C(z),\nonumber \end{eqnarray} where
$C(z)=(\lambda_{D}2^{7/2})/\sqrt{3 \pi}(\psi(z)\int_{D}\psi(y)dy) ^{2}$. Here
$\lambda_{D}$ is the first eigenvalue of the Dirichlet Laplacian ${1/2}\Delta$
in $D$, and $\psi $ is the eigenfunction corresponding to $\lambda_{D}$ .
We also study lifetime asymptotics of Brownian-time Brownian motion (BTBM),
$Z^{1}_{t}=z+X(|Y(t)|)$, where $X_{t}$ and $Y_{t}$ are independent
one-dimensional Brownian motions.
http://arXiv.org/abs/math/0603637
http://front.math.ucdavis.edu/math.PR/0603637
(alternate)
Author(s): Leonard N. Choup
Abstract: We derive expansions of the Hermite and Laguerre kernels at the edge of the
spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n
Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n kernel
expansions, we prove an Edgeworth type theorem for the largest eigenvalue
distribution function of GUEn and LUEn. In our Edgeworth expansion, the
correction terms are expressed in terms of the same Painleve II function
appearing in the leading term, i.e. in the Tracy-Widom distribution. We
conclude with a brief discussion of the universality of these results.
http://arXiv.org/abs/math/0603639
http://front.math.ucdavis.edu/math.PR/0603639
(alternate)
Author(s): Alexander E. Holroyd
Abstract: In the modified bootstrap percolation model, sites in the cube {1,...,L}^d
are initially declared active independently with probability p. At subsequent
steps, an inactive site becomes active if it has at least one active nearest
neighbour in each of the d dimensions, while an active site remains active
forever. We study the probability that the entire cube is eventually active.
For all d>=2 we prove that as L\to\infty and p\to 0 simultaneously, this
probability converges to 1 if L=exp^{d-1} (lambda+epsilon)/p, and converges to
0 if L=exp^{d-1} (lambda-epsilon)/p, for any epsilon>0. Here exp^n denotes the
n-th iterate of the exponential function, and the threshold lambda equals
pi^2/6 for all d.
http://arXiv.org/abs/math/0603645
http://front.math.ucdavis.edu/math.PR/0603645
(alternate)
Author(s): Oliver Johnson
Abstract: We prove that the Poisson distribution maximises entropy in the class of
ultra-log-concave distributions, extending a result of Harremo\"{e}s. The proof
uses ideas concerning log-concavity, and a semigroup action involving adding
Poisson variables and thinning. We go on to show that the entropy is a concave
function along this semigroup.
http://arXiv.org/abs/math/0603647
http://front.math.ucdavis.edu/math.PR/0603647
(alternate)
Author(s): M. D. Jara and C. Landim
Abstract: For a sequence of i.i.d. random variables $\{\xi_x : x\in \bb Z\}$ bounded
above and below by strictly positive finite constants, consider the
nearest-neighbor one-dimensional simple exclusion process in which a particle
at $x$ (resp. $x+1$) jumps to $x+1$ (resp. $x$) at rate $\xi_x$. We examine a
quenched nonequilibrium central limit theorem for the position of a tagged
particle in the exclusion process with bond disorder $\{\xi_x : x\in \bb Z\}$.
We prove that the position of the tagged particle converges under diffusive
scaling to a Gaussian process if the other particles are initially distributed
according to a Bernoulli product measure associated to a smooth profile
$\rho_0:\bb R\to [0,1]$.
http://arXiv.org/abs/math/0603653
http://front.math.ucdavis.edu/math.PR/0603653
(alternate)
Author(s): Saul Jacka and Abdel Berkaoui
Abstract: We consider the problem of decomposing monetary risk in the presence of a
fully traded market in {\it some} risks. We show that a mark-to-market approach
to pricing leads to such a decomposition if the risk measure is time-consistent
in the sense of Delbaen.
http://arXiv.org/abs/math/0603041
http://front.math.ucdavis.edu/math.PR/0603041
(alternate)
Author(s): M. Hairer and A. Ohashi
Abstract: We develop a theory of ergodicity for a class of random dynamical systems
where the driving noise is not white. The two main tools of our analysis are
the strong Feller property and topological irreducibility, introduced in this
work for a class of non-Markovian systems. They allow us to obtain a criteria
for ergodicity which is similar in nature to the Doob-Khas'minskii theorem.
The second part of this article shows how it is possible to apply these
results to the case of stochastic differential equations driven by fractional
Brownian motion. It follows that under a non-degeneracy condition on the noise,
such equations admit a unique adapted stationary solution.
http://arXiv.org/abs/math/0603658
http://front.math.ucdavis.edu/math.PR/0603658
(alternate)
Author(s): E. Arias-Castro and D. L. Donoho and X. Huo and C. A. Tovey
Abstract: Correction for Adv. in Appl. Probab. 37, no. 3 (2005), 571-603
http://arXiv.org/abs/math/0603673
http://front.math.ucdavis.edu/math.PR/0603673
(alternate)
Author(s): Stefan Adams and Wolfgang K\"onig
Abstract: Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some
non-degenerate initial measure 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 $1,...,N$. Such systems play an important role in quantum
physics in the description of Boson systems at positive temperature $1/\beta$.
In this paper, we describe the large-N behaviour of the empirical path
measure (the mean of the Dirac measures in the $N$ paths) and of the mean of
the normalised occupation measures of the $N$ motions in terms of large
deviations principles. The rate functions are given as variational formulas
involving certain entropies and Fenchel-Legendre transforms. Consequences are
drawn for asymptotic independence statements and laws of large numbers.
In the special case related to quantum physics, our rate function for the
occupation measures turns out to be equal to the well-known Donsker-Varadhan
rate function for the occupation measures of one motion in the limit of
diverging time. This enables us to prove a simple formula for the large-N
asymptotic of the symmetrised trace of ${\rm e}^{-\beta \mathcal{H}_N}$, where
$\mathcal{H}_N$ is an $N$-particle Hamilton operator in a trap.
http://arXiv.org/abs/math/0603702
http://front.math.ucdavis.edu/math.PR/0603702
(alternate)
Author(s): Martin Meier
Abstract: The probabilistic type spaces in the sense of Harsanyi [Management Sci. 14
(1967/68) 159--182, 320--334, 486--502] are the prevalent models used to
describe interactive uncertainty. In this paper we examine the existence of a
universal type space when beliefs are described by finitely additive
probability measures. We find that in the category of all type spaces that
satisfy certain measurability conditions ($\kappa$-measurability, for some
fixed regular cardinal $\kappa$), there is a universal type space (i.e., a
terminal object) to which every type space can be mapped in a unique
beliefs-preserving way. However, by a probabilistic adaption of the elegant
sober-drunk example of Heifetz and Samet [Games Econom. Behav. 22 (1998)
260--273] we show that if all subsets of the spaces are required to be
measurable, then there is no universal type space.
http://arXiv.org/abs/math/0602656
http://front.math.ucdavis.edu/math.PR/0602656
(alternate)
Author(s): B\'alint Vet\H{o}
Abstract: We consider permutations of $\{1,...,n\}$ obtained by $\sqrt{nt}$ independent
applications of random stirring. In each step the same marked stirring element
is transposed with probability $1/n$ with any one of the $n$ elements.
Normalizing by $\sqrt{n}$ we describe the asymptotic distribution of the cycle
structure of these permutations, for all $t\ge0$, as $n\to\infty$.
http://arXiv.org/abs/math/0603044
http://front.math.ucdavis.edu/math.PR/0603044
(alternate)
Author(s): Pascal Moyal
Abstract: In this paper we solve a particular stochastic recursion in the stationary
ergodic framework, and propose some applications of this result to the study of
regenerativity (that is, finiteness of busy cycles) and stationarity of some
queueing systems: pure delay systems, in which all customers are immediately
served, and queues with impatient customers. In this latter case under the FIFO
discipline, we prove as well the existence of a stationary workload on an
enlarged probability space.
http://arXiv.org/abs/math/0603709
http://front.math.ucdavis.edu/math.PR/0603709
(alternate)
Author(s): Enkelejd Hashorva
Abstract: Let (X_n,Y_n), n\ge 1 be bivariate random claim sizes with common
distribution function F and let N(t), t \ge 0 be a stochastic process which
counts the number of claims that occur in the time interval [0,t], t\ge 0. In
this paper we derive the joint asymptotic distribution of randomly indexed
order statistics of the random sample
(X_1,Y_1),(X_2,Y_2),...,(X_{N(t)},Y_{N(t)}) which is then used to obtain
asymptotic representations for the joint distribution of two generalised
largest claims reinsurance treaties available under specific insurance
settings. As a by-product we obtain a stochastic representation of a
m-dimensional Lambda-extremal variate in terms of iid unit exponential random
variables.
http://arXiv.org/abs/math/0603719
http://front.math.ucdavis.edu/math.PR/0603719
(alternate)
Author(s): Scott Zrebiec
Abstract: We consider a class of Gaussian random holomorphic functions, whose expected
zero set is uniformly distributed over $\C^n $. This class is unique (up to
multiplication by a non zero holomorphic function), and is closely related to a
Gaussian field over a Hilbert space of holomorphic functions on the reduced
Heisenberg group. For a fixed random function of this class, we show that the
probability that there are no zeros in a ball of large radius, is less than
$e^{-c_1 r^{2n+2}}$, and is also greater than $e^{-c_2 r^{2n+2}}$. Enroute to
this result we also compute probability estimates for the event that a random
function's unintegrated counting function deviates significantly from its mean.
http://arXiv.org/abs/math/0603696
http://front.math.ucdavis.edu/math.CV/0603696
(alternate)
Author(s): Rui Dong and Alexander Gnedin and Jim Pitman
Abstract: Kingman derived the Ewens sampling formula for random partitions describing
the genetic variation in a neutral mutation model defined by a Poisson process
of mutations along lines of descent governed by a simple coalescent process,
and observed that similar methods could be applied to more complex models.
M{\"o}hle described the recursion which determines the generalization of the
Ewens sampling formula in the situation when the lines of descent are governed
by a $\Lambda$-coalescent, which allows multiple mergers. Here we show that the
basic integral representation of transition rates for the $\Lambda$-coalescent
is forced by sampling consistency under more general assumptions on the
coalescent process. Exploiting an analogy with the theory of regenerative
partition structures, we provide various characterizations of the associated
partition structures in terms of discrete-time Markov chains.
http://arXiv.org/abs/math/0603745
http://front.math.ucdavis.edu/math.PR/0603745
(alternate)
Author(s): Vincent Lemaire (LAMA)
Abstract: The aim of this paper is to study the behavior of the weighted empirical
measures of the decreasing step Euler scheme of a one-dimensional diffusion
process having multiple invariant measures. This situation can occur when the
drift and the diffusion coefficient are vanish simultaneously. As a first step,
we give a brief description of the Feller's classification of the
one-dimensional process. We recall the concept of attractive and repulsive
boundary point and introduce the concept of strongly repulsive point. That
allows us to establish a classification of the ergodic behavior of the
diffusion. We conclude this section by giving necessary and sufficient
conditions on the nature of boundary points in terms of Lyapunov functions. In
the second section we use this characterization to study the decreasing step
Euler scheme. We give also an numerical example in higher dimension.
http://arXiv.org/abs/math/0604021
http://front.math.ucdavis.edu/math.PR/0604021
(alternate)
Author(s): Nicolas Champagnat (WIAS) and Sylvie M\'{e}l\'{e}ard (MODAL'X and FESE)
Abstract: The interplay between space and evolution is an important issue in population
dynamics, that is in particular crucial in the emergence of polymorphism and
spatial patterns. Recently, biological studies suggest that invasion and
evolution are closely related. Here we model the interplay between space and
evolution starting with an individual-based approach and show the important
role of parameter scalings on clustering and invasion. We consider a stochastic
discrete model with birth, death, competition, mutation and spatial diffusion,
where all the parameters may depend both on the position and on the trait of
individuals. The spatial motion is driven by a reflected diffusion in a bounded
domain. The interaction is modelled as a trait competition between individuals
within a given spatial interaction range. First, we give an algorithmic
construction of the process. Next, we obtain large population approximations,
as weak solutions of nonlinear reaction-diffusion equations with Neumann's
boundary conditions. As the spatial interaction range is fixed, the
nonlinearity is nonlocal. Then, we make the interaction range decrease to zero
and prove the convergence to spatially localized nonlinear reaction-diffusion
equations, with Neumann's boundary conditions. Finally, simulations based on
the microscopic individual-based model are given, illustrating the strong
effects of the spatial interaction range on the emergence of spatial and
phenotypic diversity (clustering and polymorphism) and on the interplay between
invasion and evolution. The simulations focus on the qualitative differences
between local and nonlocal interactions.
http://arXiv.org/abs/math/0604041
http://front.math.ucdavis.edu/math.PR/0604041
(alternate)
Author(s): David White
Abstract: We construct a stochastic process whose drift is a function of the process's
local time at a reflecting barrier. The process arose as a model of the
interactions of a Brownian particle and an inert particle in \citep{knight:01}.
Interesting asymptotic results are obtained for two different arrangements of
inert particles and Brownian particles. A version of the process in $\Re^d$ is
also constructed.
http://arXiv.org/abs/math/0604052
http://front.math.ucdavis.edu/math.PR/0604052
(alternate)
Author(s): Ross G. Pinsky
Abstract: Let the random variable $Z_{n,k}$ denote the number of increasing
subsequences of length $k$ in a random permutation from $S_n$, the symmetric
group of permutations of $\{1,...,n\}$. In a recent paper
(http://front.math.ucdavis.edu/math.PR/0407353) we showed that the weak law of
large numbers holds for $Z_{n,k_n}$ if $k_n=o(n^\frac25)$; that is, $$
\lim_{n\to\infty}\frac{Z_{n,k_n}} {EZ_{n,k_n}}=1, \text{in probability}. $$ The
method of proof employed there used the second moment method and demonstrated
that this method cannot work if the condition $k_n=o(n^\frac25)$ does not hold.
It follows from results concerning the longest increasing subsequence of a
random permutation that the law of large numbers cannot hold for $Z_{n,k_n}$ if
$k_n\ge cn^\frac12$, with $c>2$. Presumably there is a critical exponent $l_0$
such that the law of large numbers holds if $k_n=O(n^l)$, with $l0$, for some $l>l_0$.
Several phase transitions concerning increasing subsequences occur at
$l=\frac12$, and these would suggest that $l_0=\frac12$. However, in this
paper, we show that the law of large numbers fails for $Z_{n,k_n}$ if
$\limsup_{n\to\infty}\frac{k_n}{n^\frac49}=\infty$. Thus the critical exponent,
if it exists, must satisfy $l_0\in[\frac25,\frac49]$.
http://arXiv.org/abs/math/0604067
http://front.math.ucdavis.edu/math.PR/0604067
(alternate)
Author(s): C. Kuelske and E. Orlandi
Abstract: We prove a finite volume lower bound of the order of the squareroot of log N
on the delocalization of a disordered continuous spin model (resp. effective
interface model) in d = 2 in a box of size N . The interaction is assumed to be
massless, possibly anharmonic and dominated from above by a Gaussian. Disorder
is entering via a linear source term. For this model delocalization with the
same rate is proved to take place already without disorder, so our proof shows
that randomness will only enhance fluctuations.
http://arXiv.org/abs/math/0604068
http://front.math.ucdavis.edu/math.PR/0604068
(alternate)
Author(s): Alexis Devulder (PMA)
Abstract: We consider a one-dimensional diffusion process in a drifted Brownian
potential. We are interested in the maximum of its local time, and study its
almost sure asymptotic behaviour, which is proved to be different from the
behaviour of the maximum local time of the transient random walk in random
environment.
http://arXiv.org/abs/math/0604078
http://front.math.ucdavis.edu/math.PR/0604078
(alternate)
Author(s): Dmitry Panchenko
Abstract: We develop the cavity method for the spherical Sherrington-Kirkpatrick model
at high temperature and small external field. As one application, we carry out
the second moment computations for the overlap and the magnetization.
http://arXiv.org/abs/math/0604081
http://front.math.ucdavis.edu/math.PR/0604081
(alternate)
Author(s): Dmitry Panchenko and Michel Talagrand
Abstract: In order to study certain questions concerning the distribution of the
overlap in Sherrington-Kirkpatrick type models, such as the chaos and
ultrametricity problems, it seems natural to study the free energy of multiple
systems with constrained overlaps. One can write analogues of Guerra's replica
symmetry breaking bound for such systems but it is not at all obvious how to
choose informative functional order parameters in these bounds. We were able to
make some progress for spherical pure $p$-spin SK models where many
computations can be made explicitly. For pure 2-spin model we prove
ultrametricity and chaos in an external field. For the pure $p$-spin model for
even $p>4$ without an external field we describe two possible values of the
overlap of two systems at different temperatures. We also prove a somewhat
unexpected result which shows that in the 2-spin model the support of the joint
overlap distribution is not always witnessed at the level of the free energy
and,for example, ultrametricity holds only in a weak sense.
http://arXiv.org/abs/math/0604082
http://front.math.ucdavis.edu/math.PR/0604082
(alternate)
Author(s): Guangyue Han and Brian Marcus
Abstract: Consider a hidden Markov chain obtained as the observation process of an
ordinary Markov chain corrupted by noise. Zuk, et. al. [13], [14] showed how,
in principle, one can explicitly compute the derivatives of the entropy rate of
at extreme values of the noise. Namely, they showed that the derivatives of
standard upper approximations to the entropy rate actually stabilize at an
explicit finite time. We generalize this result to a natural class of hidden
Markov chains called ``Black Holes.'' We also discuss in depth special cases of
binary Markov chains observed in binary symmetric noise, and give an abstract
formula for the first derivative in terms of a measure on the simplex due to
Blackwell.
http://arXiv.org/abs/cs/0603059
http://front.math.ucdavis.edu/cs.IT/0603059
(alternate)
Author(s): Thomas Duquesne
Abstract: This paper is a detailled study of the coding of real trees by real valued
functions that is motivated by probabilistic problems related to continuum
random trees. Indeed it is known since the works of Aldous (1993) and Le Gall
(1991) that a continuous non-negative function $h$ on $[0,1]$ such that
$h(0)=0$ can be seen as the contour process of a compact real tree. This
particular coding of a compact real tree provides additional structures, namely
a root that is the vertex corresponding to $0\in [0,1]$, a linear order
inherited from the usual order on $[0,1]$ and a measure induced by the Lebesgue
measure on $[0,1]$; of course, the root, the linear order and the measure
obtained by such a coding have to satisfy some compatibility conditions. In
this paper, we prove that any compact real tree equipped with a root, a linear
order and a measure that are compatible can be encoded by a non-negative
function $h$ defined on a finite interval $[0, M]$, that is assumed to be
left-continuous with right-limit, without positive jump and such that
$h(0+)=h(0)=0$. Moreover, this function is unique if we assume that the
exploration of the tree induced by such a coding backtracks as less as
possible. We also prove that a measure-change on the tree corresponds to a
re-parametrization of the coding function. In addition, we describe several
path-properties of the coding function in terms of the metric properties of the
real tree.
http://arXiv.org/abs/math/0604106
http://front.math.ucdavis.edu/math.PR/0604106
(alternate)
Author(s): J.C. Pardo
Abstract: We establish integral tests and laws of the iterated logarithm for the upper
envelope of the future infimum of positive self-similar Markov processes and
for increasing self-similar Markov processes at 0 and infinity. Our proofs are
based on the Lamperti representation and time reversal arguments due to
Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for
the future infimum of Bessel processes due to Khoshnevisan et al. [11].
http://arXiv.org/abs/math/0604110
http://front.math.ucdavis.edu/math.PR/0604110
(alternate)
Author(s): Lancelot F. James
Abstract: In recent years there have been many proposals as flexible alternatives to
Gaussian based continuous time stochastic volatility models. A great deal of
these models employ positive L\'evy processes. Among these are the attractive
non-Gaussian positive Ornstein-Uhlenbeck (OU) processes proposed by
Barndorff-Nielsen and Shephard (BNS) in a series of papers. One current problem
of these approaches is the unavailability of a tractable likelihood based
statistical analysis for the returns of financial assets. This paper, while
focusing on the BNS models, develops general theory for the implementation of
statistical inference for a host of models. Specifically we show how to reduce
the infinite-dimensional process based models to finite, albeit high,
dimensional ones. Inference can then be based on Monte Carlo methods. As
highlights, specific to BNS we show that an OU process driven by an infinite
activity Gamma process, that is an OU-$\Gamma$, exhibits unique features which
allows one to exactly sample from relevant joint distributions. We show that
this is a consequence of the OU structure and the unique calculus of Gamma and
Dirichlet processes. Owing to another connection between Gamma/Dirichlet
processes and the theory of Generalized Gamma Convolutions (GGC) we identify a
large class of models, we call (FGGC), where one can perfectly sample marginal
distributions relevant to option pricing and Monte Carlo likelihood analysis.
This involves a curious result, we establish as Theorem 6.1. We also discuss
analytic techniques and candidate densities for Monte-Carlo procedures which
can be applied to more general
http://arXiv.org/abs/math/0604086
http://front.math.ucdavis.edu/math.ST/0604086
(alternate)
Author(s): N.V. Krylov
Abstract: The maximum principle for SPDEs is established in multidimensional $C^{1}$
domains. An application is given to proving the H\"older continuity up to the
boundary of solutions of one-dimensional SPDEs.
http://arXiv.org/abs/math/0604125
http://front.math.ucdavis.edu/math.PR/0604125
(alternate)
Author(s): kais Hamza and Fima C. Klebaner
Abstract: We construct a family of non-Gaussian martingales the marginals of which are
all Gaussian. We give the predictable quadratic variation of these processes
and show they do not have continuous paths. These processes are Markovian and
inhomogeneous in time, and we give their infinitesimal generators. Within this
family we find a class of piecewise deterministic pure jump processes and
describe the laws of jumps and times between the jumps.
http://arXiv.org/abs/math/0604127
http://front.math.ucdavis.edu/math.PR/0604127
(alternate)
Author(s): V.P.Kurenok
Abstract: Using the method of Krylov's estimates, we prove the existence of weak
solutions of stochastic differential equations driven by purely discontinuous
Levy processes satisfying an additional assumption. The diffusion coefficient
is assumed to be one and the time-dependent drift is measurable and bounded.
http://arXiv.org/abs/math/0604136
http://front.math.ucdavis.edu/math.PR/0604136
(alternate)
Author(s): Svante Janson
Abstract: An example is given which shows that, in general, conditioned Galton-Watson
trees cannot be obtained by adding vertices one by one, as has been shown in a
special case by Luczak and Winkler.
http://arXiv.org/abs/math/0604141
http://front.math.ucdavis.edu/math.PR/0604141
(alternate)
Author(s): S Satheesh and E Sandhya
Abstract: We discuss semi-selfdecomposable laws in the minimum scheme and characterize
them using an autoregressive model. Semi-Pareto and semi-Weibull laws of Pillai
(1991) are shown to be semi-selfdecomposable in this scheme. Methods for
deriving this class of laws are then attempted from the angle of randomization.
Finally, discrete analogues of these results are also considered.
http://arXiv.org/abs/math/0604146
http://front.math.ucdavis.edu/math.PR/0604146
(alternate)
Author(s): Manuel Lladser
Abstract: Uniform asymptotic formulae for arrays of complex numbers of the form
$(f_{r,s})$, with $r$ and $s$ nonnegative integers, are provided as $r$ and $s$
converge to infinity at a comparable rate. Our analysis is restricted to the
case in which the generating function $F(z,w):=\sum f_{r,s} z^r w^s$ is
meromorphic in a neighborhood of the origin. We provide uniform asymptotic
formulae for the coefficients $f_{r,s}$ along directions in the $(r,s)$-lattice
determined by regular points of the singular variety of $F$. Our main result
derives from the analysis of a one dimensional parameter-varying integral
describing the asymptotic behavior of $f_{r,s}$. We specifically consider the
case in which the phase term of this integral has a unique stationary point,
however, allowing the possibility that one or more stationary points of the
amplitude term coalesce with this. Our results find direct application in
certain problems associated to the Lagrange inversion formula as well as
bivariate generating functions of the form $v(z)/(1-w\cdot u(z))$.
http://arXiv.org/abs/math/0604152
http://front.math.ucdavis.edu/math.CO/0604152
(alternate)
Author(s): Boaz Nadler
Abstract: {\em Ziggurat} and {\em Monty Python} are two fast and elegant methods
proposed by Marsaglia and Tsang to transform uniform random variables to random
variables with normal, exponential and other common probability distributions.
While the proposed methods are theoretically correct, we show that there are
various design flaws in the uniform pseudo random number generators (PRNG's) of
their published implementations for both the normal and Gamma distributions
\cite{Ziggurat,{Gamma},Monty}. These flaws lead to non-uniformity of the
resulting pseudo-random numbers and consequently to noticeable deviations of
their outputs from the required distributions. In addition, we show that the
underlying uniform PRNG of the published implementation of Matlab's
\texttt{randn}, which is also based on the Ziggurat method, is not uniformly
distributed with correlations between consecutive pairs. Also, we show that the
simple linear initialization of the registers in matlab's \texttt{randn} may
lead to non-trivial correlations between output sequences initialized with
different (related or even random unrelated) seeds. These, in turn, may lead to
erroneous results for stochastic simulations.
http://arXiv.org/abs/math/0603058
http://front.math.ucdavis.edu/math.ST/0603058
(alternate)
Author(s): Vyacheslav M. Abramov
Abstract: The paper studies a single-server queueing system with autonomous service and
$\ell$ priority classes. Arrival and departure processes are defined by marked
point processes. There are $\ell$ buffers corresponding to priority classes,
and upon arrival a unit of the $k$th priority class occupies the place in the
$k$th buffer. Let $N^{(k)}$, $k=1,2,...,\ell$ denote the quota for the total
$k$th buffer content. The values $N^{(k)}$ are assumed to be large, and
queueing systems both with finite and infinite buffers are studied. In the case
of system with finite buffers, the values $N^{(k)}$ characterize buffer
capacities.
The paper discusses a circle of problems related to optimization of
performance measures associated with overflowing the quota of buffer contents.
Our approach to this problem is new, and presentation of our results is simple
and clear for real applications.
http://arXiv.org/abs/math/0604182
http://front.math.ucdavis.edu/math.PR/0604182
(alternate)
Author(s): Nathanael Berestycki and Rick Durrett
Abstract: This investigation is motivated by a result we proved recently for the random
transposition random walk: the distance from the starting point of the walk has
a phase transition from a linear regime to a sublinear regime at time $n/2$.
Here, we study three new examples. It is trivial that the distance for random
walk on the hypercube is smooth and is given by one simple formula. In the case
of random adjacent transpositions, we find that there is no phase transition
even though the distance has different scalings in three different regimes. In
the case of a random 3-regular graph, there is a phase transition from linear
growth to a constant equal to the diameter of the graph, at time $3\log_2 n$.
http://arXiv.org/abs/math/0604188
http://front.math.ucdavis.edu/math.PR/0604188
(alternate)
Author(s): Ben Hambly and James B. Martin
Abstract: We consider last-passage percolation models in two dimensions, in which the
underlying weight distribution has a heavy tail of index alpha<2. We prove
scaling laws and asymptotic distributions, both for the passage times and for
the shape of optimal paths; these are expressed in terms of a family (indexed
by alpha) of "continuous last-passage percolation" models in the unit square.
In the extreme case alpha=0 (corresponding to a distribution with slowly
varying tail) the asymptotic distribution of the optimal path can be
represented by a random self-similar measure on [0,1], whose multifractal
spectrum we compute. By extending the continuous last-passage percolation model
to R^2 we obtain a heavy-tailed analogue of the Airy process, representing the
limit of appropriately scaled vectors of passage times to different points in
the plane. We give corresponding results for a directed percolation problem
based on alpha-stable Levy processes, and indicate extensions of the results to
higher dimensions.
http://arXiv.org/abs/math/0604189
http://front.math.ucdavis.edu/math.PR/0604189
(alternate)
Author(s): Masahito Hayashi
Abstract: We formulate two types of extensions of the large deviation theory initiated
by Bahadur in a non-regular setting. One can be regarded as a bound of the
point estimation, the other can be regarded as the limit of a bound of the
interval estimation. Both coincide in the regular case, but do not necessarily
coincide in a non-regular case. Using the limits of relative R\'{e}nyi
entropies, we derive their upper bounds and give a necessary and sufficient
condition for the coincidence of the two upper bounds. We also discuss the
attainability of these two bounds in several non-regular location shift
families.
http://arXiv.org/abs/math/0604197
http://front.math.ucdavis.edu/math.PR/0604197
(alternate)
Author(s): V. Limic and P. Tarres
Abstract: The goal is to show that an edge reinforced random walk on a graph of bounded
degree, with reinforcement {\em weight function} $W$ taken from a general class
of reciprocally summable reinforcement weight functions, traverses a random
{\em attracting} edge at all large times.
The statement of the main theorem is very close to settling the original
conjecture of Sellke (1994). An important corollary of this main result says
that if $W$ is reciprocally summable and nondecreasing, the attracting edge
exists on any graph of bounded degree, with probability 1. Another corollary is
the main theorem of Limic (2003) where the class of weights was restricted to
reciprocally summable powers.
The proof uses martingale and other techniques developed by the authors in
separate studies of edge and vertex reinforced walks (Limic (2003), Tarr\`es
(2004)), and of nonconvergence properties of stochastic algorithms towards
unstable equilibrium points of the associated deterministic dynamics, Tarr\`es
(2000).
http://arXiv.org/abs/math/0604200
http://front.math.ucdavis.edu/math.PR/0604200
(alternate)
Author(s): Frederik S Herzberg
Abstract: Consider an $L^1$-continuous functional $\ell$ on the vector space of
polynomials of Brownian motion at given times, suppose $\ell $ commutes with
the quadratic variation in a natural sense, and consider a finite set of
polynomials of Brownian motion at rational times, $p_1(\vec b),...,p_m,(\vec
b)$, mapping the Wiener space to $\mathbb{R}$.
Similarly to the moment problem for a finite-dimensional space of
polynomials, we give sufficient conditions under which $\ell$ can be written in
the form $\int \cdot d\mu$ for some finite measure $\mu$ on the Wiener space
such that $\mu$-almost surely, all the random variables $p_1(\vec
b),...,p_m,(\vec b)$ are nonnegative.
http://arXiv.org/abs/math/0604211
http://front.math.ucdavis.edu/math.PR/0604211
(alternate)
Author(s): Christian H. Gromoll (STANFORD-MATHS) and Philippe Robert (INRIA Rocquencourt), Bert Zwart (TUE)
Abstract: We investigate a processor sharing queue with renewal arrivals and generally
distributed service times. Impatient jobs may abandon the queue, or renege,
before completing service. The corresponding stochastic processes are
represented by measure valued Markov processes on R^2\_+. A scaling procedure
that gives rise to a fluid model with a nontrivial, yet tractable steady state
behavior, is presented. This fluid model model captures many essential features
of the underlying stochastic model, and it is used to analyze the impact of
impatience in processor sharing queues.
http://arXiv.org/abs/math/0604215
http://front.math.ucdavis.edu/math.PR/0604215
(alternate)
Author(s): R.M. Haralick and A.D. Miasnikov and A.G. Myasnikov
Abstract: In this paper we discuss several heuristic strategies which allow one to
solve the Whitehead's minimization problem much faster (on most inputs) than
the classical Whitehead algorithm. The mere fact that these strategies work in
practice leads to several interesting mathematical conjectures. In particular,
we conjecture that the length of most non-minimal elements in a free group can
be reduced by a Nielsen automorphism which can be identified by inspecting the
structure of the corresponding Whitehead Graph.
http://arXiv.org/abs/math/0604204
http://front.math.ucdavis.edu/math.GR/0604204
(alternate)
Author(s): Andrea Montanari and Guilhem Semerjian
Abstract: Glassy systems are characterized by an extremely sluggish dynamics without
any simple sign of long range order. It is a debated question whether a correct
description of such phenomenon requires the emergence of a large correlation
length. We prove rigorous bounds between length and time scales implying the
growth of a properly defined length when the relaxation time increases. Our
results are valid in a rather general setting, which covers finite-dimensional
and mean field systems.
As an illustration, we discuss the Glauber (heat bath) dynamics of p-spin
glass models on random regular graphs. We present the first proof that a model
of this type undergoes a purely dynamical phase transition not accompanied by
any thermodynamic singularity.
http://arXiv.org/abs/cond-mat/0603018
http://front.math.ucdavis.edu/cond-mat/0603018
(alternate)
Author(s): A.D. Myasnikov and R.M Haralick
Abstract: The Whitehead Minimization problem is a problem of finding elements of the
minimal length in the automorphic orbit of a given element of a free group. The
classical algorithm of Whitehead that solves the problem depends exponentially
on the group rank. Moreover, it can be easily shown that exponential blowout
occurs when a word of minimal length has been reached and, therefore, is
inevitable except for some trivial cases.
In this paper we introduce a deterministic Hybrid search algorithm and its
stochastic variation for solving the Whitehead minimization problem. Both
algorithms use search heuristics that allow one to find a length-reducing
automorphism in polynomial time on most inputs and significantly improve the
reduction procedure. The stochastic version of the algorithm employs a
probabilistic system that decides in polynomial time whether or not a word is
minimal. The stochastic algorithm is very robust. It has never happened that a
non-minimal element has been claimed to be minimal.
http://arXiv.org/abs/math/0604206
http://front.math.ucdavis.edu/math.GR/0604206
(alternate)
Author(s): Michel Fliess (LIX and INRIA Futurs)
Abstract: Thanks to the nonstandard formalization of fast oscillating functions, due to
P. Cartier and Y. Perrin, an appropriate mathematical framework is derived for
new non-asymptotic estimation techniques, which do not necessitate any
statistical analysis of the noises corrupting any sensor. Various applications
are deduced for multiplicative noises, for the length of the parametric
estimation windows, and for burst errors.
http://arXiv.org/abs/cs/0603003
http://front.math.ucdavis.edu/cs.CE/0603003
(alternate)
Author(s): Itai Benjamini and Ori Gurel-Gurevich and Russell Lyons
Abstract: We show that the edges crossed by a random walk in a network form a recurrent
graph a.s. In fact, the same is true when those edges are weighted by the
number of crossings.
http://arXiv.org/abs/math/0603060
http://front.math.ucdavis.edu/math.PR/0603060
(alternate)
Author(s): Hermine Bierm\'{e} (MAP5) and Mark M. Meerschaert and Hans-Peter Scheffler
Abstract: A scalar valued random field is called operator-scaling if it satisfies a
self-similarity property for some matrix E with positive real parts of the
eigenvalues. We present a moving average and a harmonizable representation of
stable operator scaling random fields by utilizing so called E-homogeneous
functions. These fields also have stationary increments and are stochastically
continuous. In the Gaussian case critical H\"{o}lder-exponents and the
Hausdorff-dimension of the sample paths are also obtained.
http://arXiv.org/abs/math/0602664
http://front.math.ucdavis.edu/math.PR/0602664
(alternate)
Author(s): David Aldous and Russell Lyons
Abstract: We investigate unimodular random networks. Our motivations include their
characterization via reversibility of an associated random walk and their
similarities to unimodular quasi-transitive graphs. We extend various theorems
concerning random walks, percolation, spanning forests, and amenability from
the known context of unimodular quasi-transitive graphs to the more general
context of unimodular random networks. We give properties of a trace associated
to unimodular random networks with applications to stochastic comparison of
continuous-time random walk.
http://arXiv.org/abs/math/0603062
http://front.math.ucdavis.edu/math.PR/0603062
(alternate)
Author(s): J. Bouttier and P. Di Francesco and E. Guitter
Abstract: We introduce generalizations of Aldous' Brownian Continuous Random Tree as
scaling limits for multicritical models of discrete trees. These discrete
models involve trees with fine-tuned vertex-dependent weights ensuring a k-th
root singularity in their generating function. The scaling limit involves
continuous trees with branching points of order up to k+1. We derive explicit
integral representations for the average profile of this k-th order
multicritical continuous random tree, as well as for its history distributions
measuring multi-point correlations. The latter distributions involve
non-positive universal weights at the branching points together with fractional
derivative couplings. We prove universality by rederiving the same results
within a purely continuous axiomatic approach based on the resolution of a set
of consistency relations for the multi-point correlations. The average profile
is shown to obey a fractional differential equation whose solution involves
hypergeometric functions and matches the integral formula of the discrete
approach.
http://arXiv.org/abs/math-ph/0603007
http://front.math.ucdavis.edu/math-ph/0603007
(alternate)
Author(s): Nabin Kumar Jana
Abstract: In this paper we study the Random energy model - so called toy model of the
spin glass theory - where the underlying distributions are compactly supported.
We prove a general theorem on the asymptotics of free energy and obtain
formulae in several interesting cases - like uniform distribution, truncated
double exponential.
http://arXiv.org/abs/math/0602666
http://front.math.ucdavis.edu/math.PR/0602666
(alternate)
Author(s): Luiz Renato Fontes and Roberto H. Schonmann
Abstract: We study the threshold $theta geq 2$ contact process on a homogeneous tree
$T_b$ of degree $kappa = b + 1$, with infection parameter $lambda geq 0$ and
started from a product measure with density $p$. The corresponding mean-field
model displays a discontinuous transition at a critical point
$lambda_c^{MF}(kappa,theta)$ and for $lambda geq lambda_c^{MF}(kappa,theta)$ it
survives iff $p geq p_c^{MF}(kappa,theta,lambda)$, where this critical density
satisfies $0 < p_c^{MF}(kappa,theta,lambda) < 1$, $lim_{lambda to infty}
p_c^{MF}(kappa,theta,lambda) = 0$. For large $b$, we show that the process on
$T_b$ has a qualitatively similar behavior when $lambda$ is small, including
the behavior at and close to the critical point $lambda_c(T_b,theta)$. In
contrast, for large $lambda$ the behavior of the process on $T_b$ is
qualitatively distinct from that of the mean-field model in that the critical
density has $p_c(T_b,theta,infty) := lim_{lambda to infty}
p_c(T_b,theta,lambda) > 0$. We also show that $lim_{b to infty} b
lambda_c(T_b,theta) = Phi_{theta}$, where $1 < Phi_2 < Phi_3 < ...$,
$lim_{theta to infty} Phi_{theta} = infty$, and $0 < liminf_{b to infty}
b^{theta(theta-1)} p_c(T_b,theta,infty) leq limsup_{b to infty}
b^{theta/(theta-1)} p_c(T_b,theta,infty) < infty$.
http://arXiv.org/abs/math/0603109
http://front.math.ucdavis.edu/math.PR/0603109
(alternate)
Author(s): Jir\^o Akahori and Chihiro Uenishi and Kouji Yano
Abstract: In this paper a stochastic equation on compact groups in discrete negative
time is studied. This is closely related to Tsirelson's stochastic differential
equation, of which any solution is non-strong. How the group action reflects on
the set of solutions is investigated. It is applied to generalize Yor's result
and give a necessary and sufficient condition for existence of a strong
solution and for uniqueness in law.
http://arXiv.org/abs/math/0603113
http://front.math.ucdavis.edu/math.PR/0603113
(alternate)
Author(s): Thomas Richthammer
Abstract: The conservation of translation as a symmetry in two-dimensional systems with
interaction is a classical subject of statistical mechanics. Here we establish
such a result for Gibbsian systems of marked particles with two-body
interaction, where the interesting cases of singular, hard-core and
discontinuous interaction are included.
http://arXiv.org/abs/math/0603140
http://front.math.ucdavis.edu/math.PR/0603140
(alternate)
Author(s): Shalom Benaim and Peter Friz
Abstract: We consider risk-neutral returns and show how their tail asymptotics
translate directly to asymptotics of the implied volatility smile, thereby
sharpening Roger Lee's celebrated moment formula. The theory of regular
variation provides the ideal mathematical framework to formulate and prove such
results. The practical value of our formulae comes from the vast literature on
tail asymptotics and our conditions are often seen to be true by simple
inspection of known results.
http://arXiv.org/abs/math/0603146
http://front.math.ucdavis.edu/math.PR/0603146
(alternate)
Author(s): Francesco Russo (LAGA) and Gerald Trutnau (SFB 343)
Abstract: We consider a new class of random partial differential equation of parabolic
type where the stochastic term is constituted by an irregular noisy drift, not
necessarily Gaussian. We provide a suitable interpretation and we study
existence. After freezing a realization of the drift (stochastic process), we
study existence and uniqueness (in some suitable sense) of the associated
parabolic equation and we investigate probabilistic interpretation.
http://arXiv.org/abs/math/0602669
http://front.math.ucdavis.edu/math.PR/0602669
(alternate)
Author(s): Serban Teodor Belinschi
Abstract: We prove that the free additive convolution of two Borel probability measures
supported on the real line can have a component that is singular continuous
with respect to the Lebesgue measure on the real line only if one of the two
measures is a point mass. The density of the absolutely continuous part with
respect to the Lebesgue measure is shown to be analytic wherever positive and
finite. The atoms of the free additive convolution of Borel probability
measures on the real line have been described by Bercovici and Voiculescu in a
previous paper.
http://arXiv.org/abs/math/0603104
http://front.math.ucdavis.edu/math.OA/0603104
(alternate)
Author(s): Nabin Kumar Jana
Abstract: In this paper the Random Energy Model(REM) under exponential type environment
is considered which includes double exponential and Gaussian cases. Limiting
Free Energy is evaluated in these models. Limiting Gibbs' distribution is
evaluated in the double exponential case.
http://arXiv.org/abs/math/0602670
http://front.math.ucdavis.edu/math.PR/0602670
(alternate)
Author(s): Thierry De La Rue (LMRS)
Abstract: V.A. Rohlin asked in 1949 whether 2-fold mixing implies 3-fold mixing for a
stationary process indexed by Z, and the question remains open today. In 1978,
F. Ledrappier exhibited a counterexample to the 2-fold mixing implies 3-fold
mixing problem, the so-called "3-dot system", but in the context of stationary
random fields indexed by ZxZ. In this work, we first present an attempt to
adapt Ledrappier's construction to the one-dimensional case, which finally
leads to a stationary process which is 2-fold but not 3-fold mixing
conditionally to the sigma-algebra generated by some factor process. Then,
using arguments coming from the theory of joinings, we will give some strong
obstacles proving that Ledrappier's counterexample can not be fully adapted to
one-dimensional stationary processes.
http://arXiv.org/abs/math/0603154
http://front.math.ucdavis.edu/math.PR/0603154
(alternate)
Author(s): Xia Chen
Abstract: We study the upper tail behaviors of the local times of the additive stable
processes. Let $X_1(t),..., X_p(t)$ be independent, $d$-dimensional symmetric
stable processes with stable index $0<\alpha\le 2$ and consider the additive
stable process $\ol{X}(t_1,..., t_p)=X_1(t_1)+... +X_p(t_p)$. Under the
condition $d<\alpha p$, we obtain a precise form of large deviation principle
for the local time $$
\eta^x\big([0,t]^p\big)=\int_0^t...\int_0^t\delta_x\big(X_1(s_1)+...
+X_p(s_p)\big)ds_1... ds_p $$ of the multi-parameter process $\ol{X}(t_1,...,
t_p)$, and for its supremum norm
$\displaystyle\sup_{x\in\R^d}\eta^x\big([0,t]^p\big)$. Our results apply to the
law of the iterated logarithm and our approach is based on Fourier analysis,
moment computation and time exponentiation.
http://arXiv.org/abs/math/0603159
http://front.math.ucdavis.edu/math.PR/0603159
(alternate)
Author(s): David Marquez-Carreras and Carles Rovira and Samy Tindel
Abstract: This note is concerned with a diluted version of the perceptron model. We
establish a replica symmetric formula at high temperature, which is achieved by
studying the asymptotic behavior of a given spin magnetization. Our main task
will be to identify the order parameter of the system.
http://arXiv.org/abs/math/0603162
http://front.math.ucdavis.edu/math.PR/0603162
(alternate)
Author(s): Shuangquan Wang and Ali Abdi
Abstract: Let $\mathbf{H}=(h_{ij})$ and $\mathbf{G}=(g_{ij})$ be two $m\times n$,
$m\leq n$, random matrices, each with i.i.d complex zero-mean unit-variance
Gaussian entries, with correlation between any two elements given by
$\mathbb{E}[h_{ij}g_{pq}^\star]=\rho \delta_{ip}\delta_{jq}$ such that
$|\rho|<1$, where ${}^\star$ denotes the complex conjugate and $\delta_{ij}$ is
the Kronecker delta. Assume $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ are
unordered singular values of $\mathbf{H}$ and $\mathbf{G}$, respectively, and
$s$ and $r$ are randomly selected from $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$,
respectively. In this paper, exact analytical closed-form expressions are
derived for the joint probability distribution function (PDF) of
$\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ using an Itzykson-Zuber-type integral,
as well as the joint marginal PDF of $s$ and $r$, by a bi-orthogonal polynomial
technique. These PDFs are of interest in multiple-input multiple-output (MIMO)
wireless communication channels and systems.
http://arXiv.org/abs/math/0603170
http://front.math.ucdavis.edu/math.PR/0603170
(alternate)
Author(s): Cecile Monthus and Thomas Garel
Abstract: In dimension $d \geq 3$, the directed polymer in a random medium undergoes a
phase transition between a free phase and a disorder dominated phase. For the
latter, Fisher and Huse have proposed a droplet theory based on the scaling of
the free energy fluctuations $\Delta F(l) \sim l^{\theta}$. On the other hand,
in related growth models belonging to the KPZ universality class, Forrest and
Tang have found that the height-height correlation function is logarithmic at
the transition. For the directed polymer model at criticality, this translates
into logarithmic free energy fluctuations $\Delta F_{T_c}(l) \sim (\ln
l)^{\sigma}$ with $\sigma=1/2$. In this paper, we propose a droplet scaling
analysis exactly at criticality based on this logarithmic scaling. Our main
conclusion is that the typical correlation length $\xi(T)$ of the low
temperature phase, diverges as $ \ln \xi(T) \sim (- \ln (T_c-T))^{1/\sigma}
\sim (- \ln (T_c-T))^{2} $. Furthermore, the logarithmic dependence of $\Delta
F_{T_c}(l)$ leads to the conclusion that the critical temperature $T_c$
actually coincides with the explicit upper bound $T_2$ derived by Derrida and
coworkers, where $T_2$ corresponds to the temperature below which the ratio
$\bar{Z_L^2}/(\bar{Z_L})^2$ diverges exponentially in $L$. Finally, since the
Fisher-Huse droplet theory was initially introduced for the spin-glass phase,
we briefly mention the similarities and differences with the directed polymer
model. If one speculates that the free energy of droplet excitations for
spin-glasses is also logarithmic at $T_c$, one obtains a logarithmic decay for
the mean square correlation function at criticality $\bar{C^2(r)} \sim 1/(\ln r
)^{\sigma}$.
http://arXiv.org/abs/cond-mat/0603041
http://front.math.ucdavis.edu/cond-mat/0603041
(alternate)
Author(s): Raphael Cerf and Reda Juerg Messikh
Abstract: We study the behavior of the two-dimensional Ising model in a finite box at
temperatures that are below, but very close to, the critical temperature. In a
regime where the temperature approaches the critical point and, simultaneously,
the size of the box grows fast enough, we establish a large deviation principle
that proves the appearance of a round Wulff crystal
http://arXiv.org/abs/math/0603178
http://front.math.ucdavis.edu/math.PR/0603178
(alternate)
Author(s): Masayuki Kumon and Akimichi Takemura and Kei Takeuchi
Abstract: We consider strong law of large numbers (SLLN) in the framework of
game-theoretic probability of Shafer and Vovk (2001). We prove several versions
of SLLN for the case that Reality's moves are unbounded. Our game-theoretic
versions of SLLN largely correspond to standard measure-theoretic results.
However game-theoretic proofs are different from measure-theoretic ones in the
explicit consideration of various hedges. In measure-theoretic proofs existence
of moments are assumed, whereas in our game-theoretic proofs we assume
availability of various hedges to Skeptic for finite prices.
http://arXiv.org/abs/math/0603184
http://front.math.ucdavis.edu/math.PR/0603184
(alternate)
Author(s): Romain Abraham (MAPMO) and Jean-Fran\c{c}ois Delmas (CERMICS)
Abstract: We consider the fragmentation at nodes of the L\'{e}vy continuous random tree
introduced in a previous paper. In this framework we compute the asymptotic for
the number of small fragments at time $\theta$. This limit is increasing in
$\theta$ and discontinuous. In the $\alpha$-stable case the fragmentation is
self-similar with index $1/\alpha$, with $\alpha \in (1,2)$ and the results are
close to those Bertoin obtained for general self-similar fragmentations but
with an additional assumtion which is not fulfilled here.
http://arXiv.org/abs/math/0603192
http://front.math.ucdavis.edu/math.PR/0603192
(alternate)
Author(s): Jean-Fran\c{c}ois Delmas (CERMICS)
Abstract: We consider the height process of a L\'{e}vy process with no negative jumps,
and its associated continuous tree representation. Using tools developed by
Duquesne and Le Gall, we construct a fragmentation process at height, which
generalizes the fragmentation at height of stable trees given by Miermont. In
this more general framework, we recover that the dislocation measures are the
same as the dislocation measures of the fragmentation at node introduced by
Abraham and Delmas, up to a factor equal to the fragment size. We also compute
the asymptotic for the number of small fragments.
http://arXiv.org/abs/math/0603193
http://front.math.ucdavis.edu/math.PR/0603193
(alternate)
Author(s): Luiz Renato Fontes and Pierre Mathieu
Abstract: We study K-processes, which are Markov processes in a denumerable state
space, all of whose elements are stable, with the exception of a single state,
starting from which the process enters finite sets of stable states with
uniform distribution. We show how these processes arise, in a particular
instance, as scaling limits of the REM-like trap model ``at low temperature'',
and subsequently derive aging results for those models in this context.
http://arXiv.org/abs/math/0603198
http://front.math.ucdavis.edu/math.PR/0603198
(alternate)
Author(s): F. Baudoin and L. Coutin
Abstract: The goal of this paper is to define and study a notion of fractional Brownian
motion on a Lie group. We define it as at the solution of a stochastic
differential equation driven by a linear fractional Brownian motion. We show
that this process has stationary increments and satisfies a local self-similar
property. Furthermore the Lie groups for which this self-similar property is
global are characterized. Finally, we prove an integration by parts formula on
the path group space and deduce the existence of a density.
http://arXiv.org/abs/math/0603199
http://front.math.ucdavis.edu/math.PR/0603199
(alternate)
Author(s): Niels Richard Hansen
Abstract: We define the reflection of a random walk at a general barrier and derive, in
case the increments are light tailed and have negative mean, a necessary and
sufficient criterion for the global maximum of the reflected process to be
finite a.s. If it is finite a.s., we show that the tail of the distribution of
the global maximum decays exponentially fast and derive the precise rate of
decay. Finally, we discuss an example from structural biology that motivated
the interest in the reflection at a general barrier.
http://arXiv.org/abs/math/0603208
http://front.math.ucdavis.edu/math.PR/0603208
(alternate)
Author(s): Sharad Goel
Abstract: A deck of $n$ cards is shuffled by repeatedly moving the top card to one of
the bottom $k_n$ positions uniformly at random. We give upper and lower bounds
on the total variation mixing time for this shuffle as $k_n$ ranges from a
constant to $n$. We also consider a symmetric variant of this shuffle in which
at each step either the top card is randomly inserted into the bottom $k_n$
positions or a random card from the bottom $k_n$ positions is moved to the top.
For this reversible shuffle we derive bounds on the $L^2$ mixing time. Finally,
we transfer mixing time estimates for the above shuffles to the lazy top to
bottom-$k$ walks that move with probability 1/2 at each step.
http://arXiv.org/abs/math/0603209
http://front.math.ucdavis.edu/math.PR/0603209
(alternate)
Author(s): R. A. Doney and A. E. Kyprianou
Abstract: We obtain a new fluctuation identity for a general L\'{e}vy process giving a
quintuple law describing the time of first passage, the time of the last
maximum before first passage, the overshoot, the undershoot and the undershoot
of the last maximum. With the help of this identity, we revisit the results of
Kl\"{u}ppelberg, Kyprianou and Maller [Ann. Appl. Probab. 14 (2004) 1766--1801]
concerning asymptotic overshoot distribution of a particular class of L\'{e}vy
processes with semi-heavy tails and refine some of their main conclusions. In
particular, we explain how different types of first passage contribute to the
form of the asymptotic overshoot distribution established in the aforementioned
paper. Applications in insurance mathematics are noted with emphasis on the
case that the underlying L\'{e}vy process is spectrally one sided.
http://arXiv.org/abs/math/0603210
http://front.math.ucdavis.edu/math.PR/0603210
(alternate)
Author(s): Hermine Bierm\'{e} (MAP5)
Abstract: We estimate the anisotropic index of an anisotropic fractional Brownian
field. For all directions, we give a convergent estimator of the value of the
anisotropic index in this direction, based on generalized quadratic variations.
We also prove a central limit theorem. First we present a result of
identification based on the asymptotic behavior of the spectral density of a
process. Then, we define Radon transforms of the anisotropic fractional
Brownian field and prove that these processes admit a spectral density
satisfying the previous assumptions.
http://arXiv.org/abs/math/0602663
http://front.math.ucdavis.edu/math.ST/0602663
(alternate)
Author(s): Erwin Bolthausen and Nicola Kistler
Abstract: We introduce a natural nonhierarchical version of Derrida's generalized
random energy model. We prove that, in the thermodynamical limit, the free
energy is the same as that of a suitably constructed GREM.
http://arXiv.org/abs/math/0603212
http://front.math.ucdavis.edu/math.PR/0603212
(alternate)
Author(s): Antoine Lejay and Miguel Martinez
Abstract: The aim of this article is to provide a scheme for simulating diffusion
processes evolving in one-dimensional discontinuous media. This scheme does not
rely on smoothing the coefficients that appear in the infinitesimal generator
of the diffusion processes, but uses instead an exact description of the
behavior of their trajectories when they reach the points of discontinuity.
This description is supplied with the local comparison of the trajectories of
the diffusion processes with those of a skew Brownian motion.
http://arXiv.org/abs/math/0603214
http://front.math.ucdavis.edu/math.PR/0603214
(alternate)
Author(s): Guy Fayolle and Cyril Furtlehner
Abstract: This report is the foreword of a series dedicated to stochastic deformations
of curves. Problems are set in terms of exclusion processes, the ultimate goal
being to derive hydrodynamic limits for these systems after proper scalings. In
this study, solely the basic \textsc{asep} system on the torus is analyzed. The
usual sequence of empirical measures, converges in probability to a
deterministic measure, which is the unique weak solution of a Cauchy problem.
The method presents some new features, letting hope for extensions to higher
dimension. It relies on the analysis of a family of parabolic differential
operators, involving variational calculus. Namely, the variables are the values
of functions at given points, their number being possibly infinite.
http://arXiv.org/abs/math/0603215
http://front.math.ucdavis.edu/math.PR/0603215
(alternate)
Author(s): J. van den Berg and R. Brouwer and B. Vagvolgyi
Abstract: A few years ago two of us introduced, motivated by the study of certain
forest-fireprocesses, the self-destructive percolation model (abbreviated as
sdp model). A typical configuration for the sdp model with parameters p and
delta is generated in three steps: First we generate a typical configuration
for the ordinary percolation model with parameter p. Next, we make all sites in
the infinite occupied cluster vacant. Finally, each site that was already
vacant in the beginning or made vacant by the above action, becomes occupied
with probability delta (independent of the other sites).
Let theta(p, delta) be the probability that some specified vertex belongs, in
the final configuration, to an infinite occupied cluster. In our earlier paper
we stated the conjecture that, for the square lattice and other planar
lattices, the function theta has a discontinuity at points of the form (p_c,
delta), with delta sufficiently small. We also showed remarkable consequences
for the forest-fire models.
The conjecture naturally raises the question whether the function theta is
continuous outside some region of the above mentioned form. We prove that this
is indeed the case. An important ingredient in our proof is a (somewhat
stronger form of a) recent ingenious RSW-like percolation result of
Bollob\'{a}s and Riordan.
http://arXiv.org/abs/math/0603223
http://front.math.ucdavis.edu/math.PR/0603223
(alternate)
Author(s): Francesco Russo and Pierre Vallois
Abstract: This paper first summarizes the foundations of stochastic calculus via
regularization and constructs through this procedure It\^o and Stratonovich
integrals. In the second part, a survey and new results are presented in
relation with finite quadratic variation processes, Dirichlet and weak
Dirichlet processes.
http://arXiv.org/abs/math/0603224
http://front.math.ucdavis.edu/math.PR/0603224
(alternate)
Author(s): Michael Aizenman and Paul Jung
Abstract: We present general results for the contact process by a method which applies
to all transitive graphs of bounded degree, including graphs of exponential
growth. The model's infection rates are varied though a common control
parameter, for which two natural transition points are defined as: i.
$\lambda_T$, the value up to which the infection dies out exponentially fast if
introduced at a single site, and ii. $\lambda_H$, the threshold for the
existence of an invariant measure with a non-vanishing density of infected
sites. It is shown here that for all transitive graphs the two thresholds
coincide. The method, which proceeds through partial differential inequalities
for the infection density, yields also generally valid bounds on two related
critical exponents. The work extends existing results whose derivations were
restricted to either the discrete-time versions of the contact process or to
graphs with subexponential growth.
http://arXiv.org/abs/math/0603227
http://front.math.ucdavis.edu/math.PR/0603227
(alternate)
Author(s): Vincent Vargas (PMA)
Abstract: In this article, we derive strong localization results for directed polymers
in random environment. We show that at "low temperature" the polymer measure is
asymptotically concentrated at a few points of macroscopic mass (we call these
points epsilon-atoms). These results are derived assuming weak conditions on
the tail decay of the random environment.
http://arXiv.org/abs/math/0603233
http://front.math.ucdavis.edu/math.PR/0603233
(alternate)
Author(s): Paul Doukhan (LS-CREST and SAMOS) and Olivier Wintenberger (SAMOS)
Abstract: This paper is aimed at sharpen a weak invariance principle for stationary
sequences in Doukhan & Louhichi (1999). Our assumption is both beyond mixing
and the causal $\theta$-weak dependence in Dedecker and Doukhan (2003); those
authors obtained a sharp result which improves on an optimal one in Doukhan
{\it et alii} (1995) under strong mixing. We prove this result and we also
precise convergence rates under existence of moments with order $>2$ while
Doukhan & Louhichi (1999) assume a moment of order $>4$. Analogously to those
authors, we use a non-causal condition to deal with some general classes of
stationary and weakly dependent sequences. Besides the previously used $\eta$-
and $\kappa$-weak dependence conditions, we introduce a mixed condition,
$\lambda$, adapted to consider Bernoulli shifts with dependent inputs.
http://arXiv.org/abs/math/0603221
http://front.math.ucdavis.edu/math.ST/0603221
(alternate)
Author(s): Jeff Kahn and Gil Kalai
Abstract: Consider a random graph G in G(n,p) and the graph property: G contains a copy
of a specific graph H. (An example to keep in mind: H is a Hamiltonian cycle.)
Let p be the minimal value for which the expected number of copies of H' in G
is at least 1/2 for every subgraph H' of H. Let q be the value for which the
probability that G contains a copy of H is 1/2. Conjecture: q/p = O(log n).
Related conjectures for general Boolean functions, and a possible connection
with discrete isoperimetry are discussed.
http://arXiv.org/abs/math/0603218
http://front.math.ucdavis.edu/math.CO/0603218
(alternate)
Author(s): Pavel M. Bleher and Arno B.J. Kuijlaars
Abstract: We consider the double scaling limit in the random matrix ensemble with an
external source $\frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM$ defined on $n\times
n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm
a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues
of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one
interval for $0 < a < 1$. These two cases were treated in Parts I and II, where
we showed that the local eigenvalue correlations have the universal limiting
behavior known from unitary random matrix ensembles. For the critical case
$a=1$ new limiting behavior occurs which is described in terms of Pearcey
integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish
this result by applying the Deift/Zhou steepest descent method to a $3 \times
3$-matrix valued Riemann-Hilbert problem which involves the construction of a
local parametrix out of Pearcey integrals. We resolve the main technical issue
of matching the local Pearcey parametrix with a global outside parametrix by
modifying an underlying Riemann surface.
http://arXiv.org/abs/math-ph/0602064
http://front.math.ucdavis.edu/math-ph/0602064
(alternate)
Author(s): Fran\c{c}ois Delarue and St\'{e}phane Menozzi
Abstract: We propose a time-space discretization scheme for quasi-linear parabolic
PDEs. The algorithm relies on the theory of fully coupled forward--backward
SDEs, which provides an efficient probabilistic representation of this type of
equation. The derivated algorithm holds for strong solutions defined on any
interval of arbitrary length. As a bypass product, we obtain a discretization
procedure for the underlying FBSDE. In particular, our work provides an
alternative to the method described in [Douglas, Ma and Protter (1996) Ann.
Appl. Probab. 6 940--968] and weakens the regularity assumptions required in
this reference.
http://arXiv.org/abs/math/0603250
http://front.math.ucdavis.edu/math.PR/0603250
(alternate)
Author(s): Andrey Pavlov
Abstract: In the article is proved,that the complex part of the analytical continuation
of the LL(Z(x)) on the negative axis is equal to cZ(x),c=const., were Z(x) is
the odd function from the wide class of functions,L(Z(x)) is the transformation
of Laplas.
http://arXiv.org/abs/math/0603258
http://front.math.ucdavis.edu/math.PR/0603258
(alternate)
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