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Probability Abstracts 108
This document contains abstracts 7954-8212
from Jan-1-2009 to February-28-2009.
They have been mailed on Mar 3, 2009.
Author(s): Zdzis{\l}aw Brze{\'z}niak and Jerzy Zabczyk
Abstract: The paper is concerned with spatial and time regularity of solutions to
linear stochastic evolution equation perturbed by L\'evy white noise "obtained
by subordination of a Gaussian white noise". Sufficient conditions for spatial
continuity are derived. It is also shown that solutions do not have in general
\cadlag modifications. General results are applied to equations with fractional
Laplacian. Applications to Burgers stochastic equations are considered as well.
http://arxiv.org/abs/0901.0028
Author(s): Z. Brzezniak and B. Goldys
Abstract: The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative
space-dependent noise is considered for a ferromagnet filling a bounded
three-dimensional domain. We show the existence of weak martingale solutions
taking values in a sphere $\mathbb S^2$. The regularity of weak solutions is
also discussed. Some of the regularity results are new even for the
deterministic Landau-Lifshitz-Gilbert equation.
http://arxiv.org/abs/0901.0039
Author(s): Damien Bankovsky
Abstract: For a bivariate \Levy process $(\xi_t,\eta_t)_{t\geq 0}$ the generalised
Ornstein-Uhlenbeck (GOU) process is defined as \[V_t:=e^{\xi_t}(z+\int_0^t
e^{-\xi_{s-}}\ud \eta_s), t\ge0,\]where $z\in\mathbb{R}.$ We present conditions
on the characteristic triplet of $(\xi,\eta)$ which ensure certain ruin for the
GOU. We present a detailed analysis on the structure of the upper and lower
bounds and the sets of values on which the GOU is almost surely increasing, or
decreasing. This paper is the sequel to \cite{BankovskySly08}, which stated
conditions for zero probability of ruin, and completes a significant aspect of
the study of the GOU.
http://arxiv.org/abs/0901.0207
Author(s): M. Jara
Abstract: We prove density and current fluctuations for two examples of symmetric,
interacting particle systems with anomalous diffusive behavior: the zero-range
process with long jumps and the zero-range process with degenerated bond
disorder. As an application, we obtain subdiffusive behavior of a tagged
particle in a simple exclusion process with variable diffusion coefficient.
http://arxiv.org/abs/0901.0229
Author(s): Graham Brightwell and Malwina Luczak
Abstract: A causal set is a partially ordered set on a countably infinite ground-set
such that each element is above finitely many others. A natural extension of a
causal set is an enumeration of its elements which respects the order.
We bring together two different classes of random processes. In one class, we
are given a fixed causal set, and we consider random natural extensions of this
causal set: we think of the random enumeration as being generated one point at
a time. In the other class of processes, we generate a random causal set, again
working from the bottom up, adding one new maximal element at each stage.
Processes of both types can exhibit a property called order-invariance: if we
stop the process after some fixed number of steps, then, conditioned on the
structure of the causal set, every possible order of generation of its elements
is equally likely.
We develop a framework for the study of order-invariance which includes both
types of example: order-invariance is then a property of probability measures
on a certain space. Our main result is a description of the extremal
order-invariant measures.
http://arxiv.org/abs/0901.0240
Author(s): Steven P. Lalley and Xinghua Zheng
Abstract: The behavior at criticality of spatial SIR (susceptible/infected/recovered)
epidemic models in dimensions two and three is investigated. In these models,
finite populations of size N are situated at the vertices of the integer
lattice, and infectious contacts are limited to individuals at the same or at
neighboring sites. Susceptible individuals, once infected, remain contagious
for one unit of time and then recover, after which they are immune to further
infection. It is shown that the measure-valued processes associated with these
epidemics, suitably scaled, converge, in the large-N limit, either to a
standard Dawson-Watanabe process (super-Brownian motion) or to a
Dawson-Watanabe process with location-dependent killing, depending on the size
of the the initially infected set. A key element of the argument is a proof of
Adler's 1993 conjecture that the local time processes associated with branching
random walks converge to the local time density process associated with the
limiting super-Brownian motion.
http://arxiv.org/abs/0901.0246
Author(s): Andr\'e Mas (I3M)
Abstract: Let $z=\sum_{i=1}^{+\infty}x_{i}^{2}/a_{i}^{2}$ where the $x_{i}$'s are i.d.d
centered with unit variance gaussian random variables and $(a_{i})
_{i\in\mathbb{N}}$ an increasing sequence such that $\sum
_{i=1}^{+\infty}a_{i}^{-2}<+\infty$. We propose an exponential-integral
representation theorem for the gaussian small ball probability $\mathbb{P}%
(z<\varepsilon) $ when $\varepsilon\downarrow0$. We start from a result by
Meyer-Wolf, Zeitouni (1993) and Dembo, Meyer-Wolf, Zeitouni (1995) who computed
this probability by means of series. We prove that $\mathbb{P}% (z<\varepsilon)
$ belongs to a class of functions introduced by de Haan, well-known in extreme
value theory, the class Gamma, for which an explicit exponential-integral
representation is available. The converse implication holds under a mild
additional assumption. Some applications are underlined in connection with
statistical inference for random functions.
http://arxiv.org/abs/0901.0264
Author(s): H. Cossette and E. Marceau and V. Maume-Deschamps
Abstract: Following an article by Muller and Pflug, we study the adjustment coefficient
of ruin theory in a context of temporal dependency. We provide a consistent
estimator of this coefficient, and perform some simulations.
http://arxiv.org/abs/0901.0182
Author(s): Peter Harremoes
Abstract: On a compact group the Haar probability measure plays the role as uniform
distribution. The entropy and rate distortion theory for this uniform
distribution is studied. New results and simplified proofs on convergence of
convolutions on compact groups are presented and they can be formulated as
entropy increases to its maximum. Information theoretic techniques and Markov
chains play a crucial role. The rate of convergence is shown to be exponential.
The results are also formulated via rate distortion functions.
http://arxiv.org/abs/0901.0015
Author(s): Anatoly N. Kochubei
Abstract: On the space $\mathbb Q_p^n$, where $p\ne 2$ and $p$ does not divide $n$, we
construct a p-adic counterpart of spherical coordinates. As applications, a
description of homogeneous distributions on $\mathbb Q_p^n$ and a skew product
decomposition of p-adic L\'evy processes are given.
http://arxiv.org/abs/0901.0071
Author(s): Graham Brightwell and Malwina Luczak
Abstract: A causal set is a countably infinite poset in which every element is above
finitely many others; causal sets are exactly the posets that have a linear
extension with the order-type of the natural numbers -- we call such a linear
extension a {\em natural extension}. We study probability measures on the set
of natural extensions of a causal set, especially those measures having the
property of {\em order-invariance}: if we condition on the set of the bottom
$k$ elements of the natural extension, each possible ordering among these $k$
elements is equally likely. We give sufficient conditions for the existence and
uniqueness of an order-invariant measure on the set of natural extensions of a
causal set.
http://arxiv.org/abs/0901.0242
Author(s): Nizar Demni
Abstract: We define and study a multidimensional process that generalizes the
eigenvalues of matrix Jacobi processes on the one hand and whose stationary
distribution is given by the beta Jacobi ensemble on the other hand.
http://arxiv.org/abs/0901.0324
Author(s): Frederi G. Viens
Abstract: We consider a random variable X satisfying almost-sure conditions involving
G:= where DX is X's Malliavin derivative and L^{-1} is the
inverse Ornstein-Uhlenbeck operator. A lower- (resp. upper-) bound condition on
G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail
P[X>z]. Bounds of other natures are also given. A key ingredient is the use of
Stein's lemma, including the explicit form of the solution of Stein's equation
relative to the function 1_{x>z}, and its relation to G. Another set of
comparable results is established, without the use of Stein's lemma, using
instead a formula for the density of a random variable based on G, recently
devised by the author and Ivan Nourdin. As an application, via a Mehler-type
formula for G, we show that the Brownian polymer in a Gaussian environment
which is white-noise in time and positively correlated in space has deviations
of Gaussian type and a fluctuation exponent \chi=1/2. We also show this
exponent remains 1/2 after a non-linear transformation of the polymer's
Hamiltonian.
http://arxiv.org/abs/0901.0383
Author(s): Theo van Uem
Abstract: We obtain expected number of arrivals, probability of arrival, absorption
probabilities and expected time before absorption for a general discrete random
walk with variable absorbing probabilities on a finite interval using Fibonacci
numbers
http://arxiv.org/abs/0901.0469
Author(s): Nicholas Crawford and Dmitry Ioffe
Abstract: Recently, a random current representation for transverse field Ising models
has been introduced in \cite{ILN}. This representation is a space-time version
of the classical random current representation exploited by Aizenman et. al.
%It is a space-time version of the classical random current representation
\cite{Ai82, ABF, AF}. In this paper we formulate and prove corresponding
space-time versions of the classical switching lemma and show how they generate
various correlation inequalities. In particular we prove exponential decay of
truncated two-point functions at positive magnetic fields in $\sfz$-direction
and address the issue of the sharpness of phase transition.
http://arxiv.org/abs/0812.4834
Author(s): Tomas Caraballo and Jinqiao Duan and Kening Lu and Bjorn Schmalfuss
Abstract: Random invariant manifolds are geometric objects useful for understanding
complex dynamics under stochastic influences. Under a nonuniform hyperbolicity
or a nonuniform exponential dichotomy condition, the existence of random
pseudo-stable and pseudo-unstable manifolds for a class of \emph{random}
partial differential equations and \emph{stochastic} partial differential
equations is shown. Unlike the invariant manifold theory for stochastic
\emph{ordinary} differential equations, random norms are not used. The result
is then applied to a nonlinear stochastic partial differential equation with
linear multiplicative noise.
http://arxiv.org/abs/0901.0382
Author(s): A. Gaudilliere and F.R. Nardi
Abstract: We consider a two type (red and blue or $R$ and $B$) particle population that
evolves on the $d$-dimensional lattice according to some reaction-diffusion
process $R+B\to 2R$ and starts with a single red particle and a density $\rho$
of blue particles. For two classes of models we give an upper bound on the
propagation velocity of the red particles front with explicit dependence on
$\rho$.
In the first class of models red and blue particles respectively evolve with
a diffusion constant $D_R=1$ and a possibly time dependent jump rate $D_B \geq
0$ -- more generally blue particles follow some independent bistochastic
process and this also includes long range random walks with drift and various
deterministic processes. We then get in all dimensions an upper bound of order
$\max(\rho,\sqrt\rho)$ that depends only on $\rho$ and $d$ and not on the
specific process followed by blue particles, in particular that does not depend
on $D_B$. We argue that for $d \geq 2$ or $\rho \geq 1$ this bound can be
optimal (in $\rho$), while for the simplest case with $d=1$ and $\rho < 1$
known as the frog model, we give a better bound of order $\rho$.
In the second class of models particles evolve with exclusion and possibly
attraction inside a large two-dimensional box with periodic boundary conditions
according to Kawasaki dynamics (that turns into simple exclusion when the
attraction is set to zero.) In a low density regime we then get an upper bound
of order $\sqrt\rho$. This proves a long-range decorrelation of dynamical
events in this low density regime.
http://arxiv.org/abs/0901.0586
Author(s): Julien Bect
Abstract: A general formulation of the Fokker-Planck-Kolmogorov (FPK) equation for
stochastic hybrid systems is presented, within the framework of Generalized
Stochastic Hybrid Systems (GSHS). The FPK equation describes the time evolution
of the probability law of the hybrid state. Our derivation is based on the
concept of mean jump intensity, which is related to both the usual stochastic
intensity (in the case of spontaneous jumps) and the notion of probability
current (in the case of forced jumps). This work unifies all previously known
instances of the FPK equation for stochastic hybrid systems, and provides GSHS
practitioners with a tool to derive the correct evolution equation for the
probability law of the state in any given example.
http://arxiv.org/abs/0901.0615
Author(s): Achim Klenke and Leonid Mytnik
Abstract: We construct a mutually catalytic branching process on a countable site space
with infinite "branching rate". The finite rate mutually catalytic model, in
which the rate of branching of one population at a site is proportional to the
mass of the other population at that site, was introduced by Dawson and Perkins
in [DP98]. We show that our model is the limit for a class of models and in
particular for the Dawson- Perkins model as the rate of branching goes to
infinity. Our process is characterized as the unique solution to a martingale
problem. We also give a characterization of the process as a weak solution of
an infinite system of stochastic integral equations driven by a Poisson noise.
http://arxiv.org/abs/0901.0623
Author(s): Pieter Trapman
Abstract: We consider long-range percolation on $\mathbb{Z}^d$ in which the measure on
the configuration of edges is a product measure and the probability that two
vertices at distance $r$ share an edge is given by $p(r) = 1-\exp[-\lambda(r)]
\in (0,1)$. Here $\lambda(r)$ is a strictly positive, non-increasing regularly
varying function. We investigate the asymptotic growth of the size of the
$k$-ball around the origin, $|\mathcal{B}_k|$, i.e. the number of vertices that
are within graph-distance $k$ of the origin, for $k \to \infty$ for different
$\lambda(r)$. We show that conditioned on the origin being in the infinite
component, non-empty classes of non-increasing regularly varying $\lambda(r)$
exist for which respectively $|\mathcal{B}_k|^{1/k} \to \infty$ almost surely,
there exist $1
http://arxiv.org/abs/0901.0661
Author(s): Philippe Flajolet and Miklos Bona
Abstract: The probability that two randomly selected phylogenetic trees of the same
size are isomorphic is found to be asymptotic to a decreasing exponential
modulated by a polynomial factor. The number of symmetrical nodes in a random
phylogenetic tree of large size obeys a limiting Gaussian distribution, in the
sense of both central and local limits. The probability that two random
phylogenetic trees have the same number of symmetries asymptotically obeys an
inverse square-root law. Precise estimates for these problems are obtained by
methods of analytic combinatorics, involving bivariate generating functions,
singularity analysis, and quasi-powers approximations.
http://arxiv.org/abs/0901.0696
Author(s): J. Harnad and A. Yu. Orlov
Abstract: Generalized convolution symmetries of integrable hierarchies of KP-Toda and
2KP-Toda type have the effect of multiplying the Fourier coefficients of the
Baker-Akhiezer function by a specified sequence of constants. The induced
action on the associated fermionic Fock space is diagonal in the standard
orthonormal base determined by occupation sites and labeled by partitions. The
coefficients in the single and double Schur function expansions of the
associated $\tau$-functions, which are the Pl\"ucker coordinates of a
decomposable element, are multiplied by the corresponding diagonal factors.
Applying such transformations to matrix integrals, we obtain new matrix models
of externally coupled type which are also KP-Toda or 2KP-Toda $\tau$-functions.
More general multiple integral representations of tau functions are similarly
obtained, as well as finite determinantal expressions for them.
http://arxiv.org/abs/0901.0323
Author(s): Hubert Lacoin
Abstract: We study the free energy of the directed polymer in random environment in
dimension 1+1 and 1+2. For dimension 1, we improve the statement of Comets and
Vargas concerning very strong disorder by giving sharp estimates on the free
energy at high temperature. In dimension 2, we prove that very strong disorder
holds at all temperatures, thus solving a long standing conjecture.
http://arxiv.org/abs/0901.0699
Author(s): Yehuda Izhakian and Zur Izhakian
Abstract: The classical probability theory supports probability measures assigning each
event with a fixed positive real value; aiming to formulate occurrences in real
life, these measures are far from being satisfactory. The main innovation of
this paper is the introduction of a new probability measure, enabling the
assignment of events with varying probabilities that are recorded by ring
elements; this measure still provides a Bayesian model, resembling the
classical probability model.
By introducing two principles for the possible variation of a probability
(also known as uncertainty, ambiguity, or imprecise probability), together with
the ``correct'' algebraic structure allowing the framing of these principles,
we present the foundations for the theory of phantom probability, generalizing
the classical probability theory in a natural way. This generalization
preserves much of the well known properties, as well as familiar distribution
functions, of the classical probability theory: moments, covariance, moment
generating functions, the low of large numbers, and the central limit theorem
are a few instances demonstrating the concept of the phantom probability
theory.
http://arxiv.org/abs/0901.0902
Author(s): Fumio Hiai and Takuho Miyamoto
Abstract: A new concept of mutual pressure is introduced for potential functions on
both continuous and discrete compound spaces via discrete micro-states of
permutations, and its relations with the usual pressure and the mutual
information are established. This paper is a continuation of the paper of Hiai
and Petz in Banach Center Publications, Vol. 78.
http://arxiv.org/abs/0901.1072
Author(s): Ramon van Handel
Abstract: The nonlinear filter for an ergodic signal observed in white noise is said to
achieve maximal accuracy if the stationary filtering error vanishes as the
signal to noise ratio diverges. We give a general characterization of the
maximal accuracy property in terms of various systems theoretic notions. When
the signal state space is a finite set explicit necessary and sufficient
conditions are obtained, while the linear Gaussian case reduces to a classic
result of Kwakernaak and Sivan (1972).
http://arxiv.org/abs/0901.1084
Author(s): Xia Chen and Wenbo Li and Michael B. Marcus and Jay Rosen
Abstract: Let $\{L^{x}_{t} ; (x,t)\in R^{1}\times R^{1}_{+}\}$ denote the local time of
Brownian motion and \[ \alpha_{t}:=\int_{-\infty}^{\infty} (L^{x}_{t})^{2} dx .
\] Let $\eta=N(0,1)$ be independent of $\alpha_{t}$. For each fixed $t$ \[
{\int_{-\infty}^{\infty} (L^{x+h}_{t}- L^{x}_{t})^{2} dx- 4ht\over h^{3/2}}
\stackrel{\mathcal{L}}{\to}({64 \over 3})^{1/2}\sqrt{\alpha_{t}} \eta, \] as
$h\rar 0$. Equivalently \[ {\int_{-\infty}^{\infty} (L^{x+1}_{t}-
L^{x}_{t})^{2} dx- 4t\over t^{3/4}} \stackrel{\mathcal{L}}{\to}({64 \over 3}
)^{1/2}\sqrt{\alpha_{1}} \eta, \] as $t\rar\infty$.
http://arxiv.org/abs/0901.1102
Author(s): Ivan Gentil (CEREMADE) and Boguslaw Zegarlinski
Abstract: In this work, we prove the existence and the exponential decay to equilibrium
of a general reversible chemical reaction-diffusion equation with same but
general diffusion. Moreover, we prove the optimal asymptotic behaviour in the
"two-by-two" case.
http://arxiv.org/abs/0901.1241
Author(s): Damiano Brigo and Giovanni Pistone
Abstract: In the present paper we discuss problems concerning evolutions of densities
related to Ito diffusions in the framework of the statistical exponential
manifold. We develop a rigorous approach to the problem, and we particularize
it to the orthogonal projection of the evolution of the density of a diffusion
process onto a finite dimensional exponential manifold. It has been shown by D.
Brigo (1996) that the projected evolution can always be interpreted as the
evolution of the density of a different diffusion process. We give also a
compactness result when the dimension of the exponential family increases, as a
first step towards a convergence result to be investigated in the future. The
infinite dimensional exponential manifold structure introduced by G. Pistone
and C. Sempi is used and some examples are given.
http://arxiv.org/abs/0901.1308
Author(s): Joan Lind and Donald E. Marshall and and Steffen Rohde
Abstract: We analyze Loewner traces driven by functions asymptotic to K\sqrt{1-t}. We
prove a stability result when K is not 4 and show that K=4 can lead to non
locally connected hulls. As a consequence, we obtain a driving term \lambda(t)
so that the hulls driven by K\lambda(t) are generated by a continuous curve for
all K > 0 with K not equal to 4 but not when K = 4, so that the space of
driving terms with continuous traces is not convex. As a byproduct, we obtain
an explicit construction of the traces driven by K\sqrt{1-t} and a conceptual
proof of the corresponding results of Kager, Nienhuis and Kadanoff,
math-ph/0309006
http://arxiv.org/abs/0901.1157
Author(s): Mark Conger and Jason Howald
Abstract: This thesis considers the effect of riffle shuffling on decks of cards,
allowing for some cards to be indistinguishable from other cards. The dual
problem of dealing a game with hands, such as bridge or poker, is also
considered. The Gilbert-Shannon-Reeds model of card shuffling is used, along
with variation distance for measuring how close to uniform a deck has become.
The surprising results are that for a deck with only two types of cards (such
as red and black), the shuffler can greatly improve the randomness of the deck
by insuring that the top and bottom cards are the same before shuffling. And in
the case of dealing cards for a game with "hands", such as bridge or poker, the
normal method of dealing cyclically around the table is very far from optimal.
In the case of a well-shuffled bridge deck, changing to another dealing method
is as good as doing 3.7 extra shuffles. How the deck is cut in poker affects
its randomness as well.
http://arxiv.org/abs/0901.1324
Author(s): Nicky Sonigo (UMPA-Ensl)
Abstract: We consider a totally asymmetric exclusion process on the positive half-line.
When particles enter in the system according to a Poisson source, Liggett has
computed all the limit distributions when the initial distribution has an
asymptotic density. In this paper we consider systems for which particles enter
at the boundary according to a complex mechanism depending on the current
configuration in a finite neighborhood of the origin. For this kind of models,
we prove a strong law of large numbers for the number of particles entered in
the system at a given time. Our main tool is a new representation of the model
as a multi-type particle system with infinitely many particle types.
http://arxiv.org/abs/0901.1364
Author(s): Kouji Yano
Abstract: Two kinds of conditionings for one-dimensional stable L\'evy processes are
discussed via $ h $-transforms of excursion measures: One is to stay positive,
and the other is to avoid the origin.
http://arxiv.org/abs/0901.1374
Author(s): Abdelhamid Hassairi and Mahdi Louati
Abstract: The aim of this paper is to study the mixture of the Riesz distribution on
symmetric matrices with respect to the multivariate Poisson distribution. We
show, in particular, that this distribution is related to the modified Bessel
function of the first kind. We also study the generated natural exponential
family. We determine the domain of the means and the variance function of this
family.
http://arxiv.org/abs/0901.1390
Author(s): Arthur Charpentier and Johan Segers
Abstract: A complete and user-friendly directory of tails of Archimedean copulas is
presented which can be used in the selection and construction of appropriate
models with desired properties. The results are synthesized in the form of a
decision tree: Given the values of some readily computable characteristics of
the Archimedean generator, the upper and lower tails of the copula are
classified into one of three classes each, one corresponding to asymptotic
dependence and the other two to asymptotic independence. For a long list of
single-parameter families, the relevant tail quantities are computed so that
the corresponding classes in the decision tree can easily be determined. In
addition, new models with tailor-made upper and lower tails can be constructed
via a number of transformation methods. The frequently occurring category of
asymptotic independence turns out to conceal a surprisingly rich variety of
tail dependence structures.
http://arxiv.org/abs/0901.1521
Author(s): J. E. Bj\"ornberg and G. R. Grimmett
Abstract: An analysis is presented of the phase transition of the quantum Ising model
with transverse field on the d-dimensional hypercubic lattice. It is shown that
there is a unique sharp transition. The value of the critical point is
calculated rigorously in one dimension. The first step is to express the
quantum Ising model in terms of a (continuous) classical Ising model in d+1
dimensions. A so-called `random-parity' representation is developed for the
latter model, similar to the random-current representation for the classical
Ising model on a discrete lattice. Certain differential inequalities are
proved. Integration of these inequalities yields the sharpness of the phase
transition, and also a number of other facts concerning the critical and
near-critical behaviour of the model under study.
http://arxiv.org/abs/0901.0328
Author(s): Yves F. Atchade
Abstract: There is a growing interest in the literature for adaptive Markov Chain Monte
Carlo methods based on sequences of random transition kernels $\{P_n\}$ where
the kernel $P_n$ is allowed to have an invariant distribution $\pi_n$ not
necessarily equal to the distribution of interest $\pi$ (target distribution).
These algorithms are designed such that as $n\to\infty$, $P_n$ converges to
$P$, a kernel that has the correct invariant distribution $\pi$. Typically, $P$
is a kernel with good convergence properties, but one that cannot be directly
implemented. It is then expected that the algorithm will inherit the good
convergence properties of $P$. The equi-energy sampler of \cite{kzw06} is an
example of this type of adaptive MCMC. We show in this paper, that the
asymptotic variance of this type of adaptive MCMC is always at least as large
as the asymptotic variance of the Markov chain with transition kernel $P$. We
also show by simulation that the difference can be substantial.
http://arxiv.org/abs/0901.1378
Author(s): Ioannis Papageorgiou (Imperial College London)
Abstract: We are interested in the Logarithmic Sobolev Inequality for the infinite
volume Gibbs measure with no quadratic interactions. We consider unbounded spin
systems on the one dimensional Lattice with interactions that go beyond the
usual strict convexity and without uniform bound on the second derivative. We
assume that the one dimensional single-site measure with boundaries satisfies
the Log-Sobolev inequality uniformly on the boundary conditions and we
determine conditions under which the Log-Sobolev Inequality can be extended to
the infinite volume Gibbs measure.
http://arxiv.org/abs/0901.1403
Author(s): Zhenting Hou and Xiangxing Kong and Dinghua Shi and Guanrong Chen and Qinggui Zhao
Abstract: In this paper, we abstract a kind of stochastic processes from evolving
processes of evolving networks, this process is called evolving network Markov
chains. Thus the degree distribution of evolving network is transformed to the
corresponding problem of evolving network Markov chains. First we investigate
the evolving network Markov chains, and get its exact formulas and obtain a
criteria to judge whether the steady degree distribution is power-law or not.
Then we apply it to evolving networks. With this method, we get a rigorous,
exact and unified solution of the steady degree distribution for evolving
networks.
http://arxiv.org/abs/0901.1418
Author(s): Ioannis Papageorgiou (Imperial College London)
Abstract: We consider unbounded spin systems on the one dimensional Lattice with
interactions that go beyond the usual strict convexity and without uniform
bound on the second derivative. We assume that the one dimensional without
interactions (boundary-free) measure satisfies the Logarithmic Sobolev
inequality and we determine conditions under which the Log-Sobolev Inequality
can be extended to the infinite volume Gibbs measure.
http://arxiv.org/abs/0901.1482
Author(s): Geoffrey Grimmett
Abstract: Correlation inequalities are presented for functionals of a ferromagnetic
Potts model with external field, using the random-cluster representation. These
results extend earlier inequalities of Ganikhodjaev--Razak and Schonmann, and
yield also GKS-type inequalities when the spin-space is taken as the set of qth
roots of unity.
http://arxiv.org/abs/0901.1625
Author(s): Karl-Theodor Sturm
Abstract: We construct the entropic measure $\mathbb{P}^\beta$ on compact manifolds of
any dimension. It is defined as the push forward of the Dirichlet process
(another random probability measure, well-known to exist on spaces of any
dimension) under the {\em conjugation map}
$$\Conj:\mathcal{P}(M)\to\mathcal{P}(M).$$ This conjugation map is a continuous
involution. It can be regarded as the canonical extension to higher dimensional
spaces of a map between probability measures on 1-dimensional spaces
characterized by the fact that the distribution functions of $\mu$ and
$\Conj(\mu)$ are inverse to each other. We also present an heuristic
interpretation of the entropic measure as
$$d\mathbb{P}^\beta(\mu)=\frac{1}{Z}\exp(-\beta\cdot {Ent} (\mu|m))\cdot
d\mathbb{P}^0(\mu).$$
http://arxiv.org/abs/0901.1815
Author(s): Giacomo Aletti and Diane Saada
Abstract: On a weakly Blackwell space we show how to define a Markov chain
approximating problem, for the target problem. The approximating problem is
proved to converge to the optimal reduced problem under different
pseudometrics. A computational example of compression of information is
discussed.
http://arxiv.org/abs/0901.1871
Author(s): Vytas Zacharovas
Abstract: The well known Erdos-Turan law states that the logarithm of an order of a
random permutation is asymptotically normally distributed. The aim of this work
is to estimate convergence rate in this theorem and also to prove analogous
result for distribution of the logarithm of an order of a random permutation on
a certain class of subsets of the symmetric group. We also study the asymptotic
behavior of the mean values of multiplicative functions on the symmetric group
and the results we obtain are of independent interest besides their application
to the investigation of the remainder term in the Erdos-Turan law.
We also study a related problem of distribution of the degree of a splitting
field of a random polynomial and obtain sharp estimates for its convergence
rate to normal law.
In research we apply both probabilistic and analytic methods. Some analytic
methods used here have their origins in the probabilistic number theory, and
some have their roots in the theory of summation of divergent series. One of
the approaches we use is to apply Tauberian type estimates for Voronoi
summability of divergent series to analyze the generating functions of the mean
values of multiplicative functions.
http://arxiv.org/abs/0901.1733
Author(s): Michael Skeide
Abstract: In these notes we tie up some loose ends in the theory of E_0-semigroups and
their classification by product systems of Hilbert modules. We explain how the
notion of cocycle conjugacy must be modified in order to see how product
systems classify E_0-semigroups. Actually, we will find two notions of cocycle
conjugacy (which for Hilbert spaces coincide) that lead to classification up to
isomorphism of product systems and up to Morita equivalence of product systems,
respectively. (In between there is also a classification up to generalized
isomorphism of product systems.)
Apart from these new results, we provide also general versions of results
known for Hilbert modules with unit vectors. In this context it is also
indispensable to review the notions of Morita equivalent product systems and
Morita equivalent Hilbert modules, adding some generalities that have not yet
been mentioned. In any case, we underline the outstanding role played by Morita
equivalence in the relation between E_0-semigroups and product systems. As
usual with Morita equivalence, the most satisfying form of the results we find
for von Neumann algebras. Some of the C*-versions of the results will depend on
countability assumptions.
Altogether, we have now a complete the theory of the classification of normal
E_0-semigroups on B^a(E) by product systems of von Neumann correspondences. We
have the same theory for the classification of strict E_0-semigroups by product
systems of C*-correspondences under countability hypotheses. In both cases, we
apply our theory to prove that a Markov semigroup admits a Hudson-Parthasarathy
dilation if and only if it is spatial.
http://arxiv.org/abs/0901.1798
Author(s): Damiano Brigo
Abstract: In this paper we consider the continuous--time nonlinear filtering problem,
which has an infinite--dimensional solution in general, as proved by
Chaleyat--Maurel and Michel. There are few examples of nonlinear systems for
which the optimal filter is finite dimensional, in particular Kalman's, Benes',
and Daum's filters. In the present paper, we construct new classes of scalar
nonlinear filtering problems admitting finite--dimensional filters. We consider
a given (nonlinear) diffusion coefficient for the state equation, a given
(nonlinear) observation function, and a given finite--dimensional exponential
family of probability densities. We construct a drift for the state equation
such that the resulting nonlinear filtering problem admits a
finite--dimensional filter evolving in the prescribed exponential family
augmented by the observaton function and its square.
http://arxiv.org/abs/0901.1952
Author(s): Kol\'eh\'e Abdoulaye Coulibaly-Pasquier (LMA)
Abstract: We generalize Brownian motion on a Riemannian manifold to the case of a
family of metrics which depends on time. Such questions are natural for
equations like the heat equation with respect to time dependent Laplacians
(inhomogeneous diffusions). In this paper we are in particular interested in
the Ricci flow which provides an intrinsic family of time dependent metrics. We
give a notion of parallel transport along this Brownian motion, and establish a
generalization of the Dohrn-Guerra or damped parallel transport, Bismut
integration by part formulas, and gradient estimate formulas. One of our main
results is a characterization of the Ricci flow in terms of the damped parallel
transport. At the end of the paper we give an intrinsic definition of the
damped parallel transport in terms of stochastic flows, and derive an intrinsic
martingale which may provide information about singularities of the flow.
http://arxiv.org/abs/0901.1999
Author(s): Samy Tindel (IECN) and Iv\'an Torrecilla (UB)
Abstract: This note is devoted to show how to push forward the algebraic integration
setting in order to treat differential systems driven by a noisy input with
H\"older regularity greater than 1/4. After recalling how to treat the case of
ordinary stochastic differential equations, we mainly focus on the case of
delay equations. A careful analysis is then performed in order to show that a
fractional Brownian motion with Hurst parameter H>1/4 fulfills the assumptions
of our abstract theorems.
http://arxiv.org/abs/0901.2010
Author(s): Bela Bollobas and Svante Janson and Oliver Riordan
Abstract: In this paper we study the component structure of random graphs with
independence between the edges. Under mild assumptions, we determine whether
there is a giant component, and find its asymptotic size when it exists. We
assume that the sequence of matrices of edge probabilities converges to an
appropriate limit object (a kernel), but only in a very weak sense, namely in
the cut metric. Our results thus generalize previous results on the phase
transition in the already very general inhomogeneous random graph model we
introduced recently, as well as related results of Bollob\'as, Borgs, Chayes
and Riordan, all of which involve considerably stronger assumptions. We also
prove corresponding results for random hypergraphs; these generalize our
results on the phase transition in inhomogeneous random graphs with clustering.
http://arxiv.org/abs/0901.2091
Author(s): H. Monien
Abstract: We derive a general expression for the Hankel determinants of a Dirichlet
series F(s) and derive the asymptotic behavior for the special case that F(s)
is the Riemann zeta function. In this case the Hankel determinant is a discrete
analogue of the Selberg integral and can be viewed as a matrix integral with
discrete measure. We briefly comment on its relation to Plancherel measures.
http://arxiv.org/abs/0901.1883
Author(s): Rayyan G. Jaber and Jeffrey G. Andrews
Abstract: In this paper, a lower bound on the capacity of wireless ad hoc erasure
networks is derived in closed form in the canonical case where $n$ nodes are
uniformly and independently distributed in the unit area square. The bound
holds almost surely and is asymptotically tight. We assume all nodes have fixed
transmit power and hence two nodes should be within a specified distance $r_n$
of each other to overcome noise. In this context, interference determines
outages, so we model each transmitter-receiver pair as an erasure channel with
a broadcast constraint, i.e. each node can transmit only one signal across all
its outgoing links. A lower bound of $\Theta(n r_n)$ for the capacity of this
class of networks is derived. If the broadcast constraint is relaxed and each
node can send distinct signals on distinct outgoing links, we show that the
gain is a function of $r_n$ and the link erasure probabilities, and is at most
a constant if the link erasure probabilities grow sufficiently large with $n$.
Finally, the case where the erasure probabilities are themselves random
variables, for example due to randomness in geometry or channels, is analyzed.
We prove somewhat surprisingly that in this setting, variability in erasure
probabilities increases network capacity.
http://arxiv.org/abs/0901.1936
Author(s): Anne De Bouard (CMAP) and Arnaud Debussche (IRMAR)
Abstract: We consider a randomly perturbed Korteweg-de Vries equation. The perturbation
is a random potential depending both on space and time, with a white noise
behavior in time, and a regular, but stationary behavior in space. We
investigate the dynamics of the soliton of the KdV equation in the presence of
this random perturbation, assuming that the amplitude of the perturbation is
small. We estimate precisely the exit time of the perturbed solution from a
neighborhood of the modulated soliton, and we obtain the modulation equations
for the soliton parameters. We moreover prove a central limit theorem for the
dispersive part of the solution, and investigate the asymptotic behavior in
time of the limit process.
http://arxiv.org/abs/0901.1965
Author(s): Klaus Fleischmann and Leonid Mytnik and Vitali Wachtel
Abstract: A H"older regularity index at given points for density states of
(alpha,1,beta)-superprocesses with alpha>1+beta is determined. It is shown that
this index is strictly greater than the optimal index of local H"older
continuity for those density states.
http://arxiv.org/abs/0901.2315
Author(s): Masoud M. Nasari
Abstract: This paper investigates weak convergence of U-statistics via approximation in
probability. The classical condition that the second moment of the kernel of
the underlying U-statistic exists is relaxed to having 4/3 moments only (modulo
a logarithmic term). Furthermore, the conditional expectation of the kernel is
only assumed to be in the domain of attraction of the normal law (instead of
the classical two-moment condition).
http://arxiv.org/abs/0901.2343
Author(s): Debasish Chatterjee and Soumik Pal
Abstract: We address stability of a class of Markovian discrete-time stochastic hybrid
systems. This class of systems is characterized by the state-space of the
system being partitioned into a safe or target set and its exterior, and the
dynamics of the system being different in each domain. We give conditions for
$L_1$-boundedness of Lyapunov functions based on certain negative drift
conditions outside the target set, together with some more minor assumptions.
We then apply our results to a wide class of randomly switched systems (or
iterated function systems), for which we give conditions for global asymptotic
stability almost surely and in $L_1$. The systems need not be time-homogeneous,
and our results apply to certain systems for which functional-analytic or
martingale-based estimates are difficult or impossible to get.
http://arxiv.org/abs/0901.2269
Author(s): Theodore P. Hill and Kent E. Morrison
Abstract: The formal mathematical theory of fair division has a rich history dating
back at least to Steinhaus in the 1940's. In recent work in this area, several
general classes of errors have appeared along with confusion about the
necessity and sufficiency of certain hypotheses. It is the purpose of this
article to correct the scientific record and to point out with concrete
examples some of the pitfalls that have led to these mistakes. These examples
may serve as guideposts for future work.
http://arxiv.org/abs/0901.2360
Author(s): Tomasz R. Bielecki and Monique Jeanblanc and Marek Rutkowski
Abstract: In the paper we study dynamics of the arbitrage prices of credit default
swaps within a hazard process model of credit risk. We derive these dynamics
without postulating that the immersion property is satisfied between some
relevant filtrations. These results are then applied so to study the problem of
replication of general defaultable claims, including some basket claims, by
means of dynamic trading of credit default swaps.
http://arxiv.org/abs/0901.2390
Author(s): Louis H. Y. Chen and Aihua Xia
Abstract: Although the study of weak convergence of superposition of point processes to
the Poisson process dates back to the work of Grigelionis in 1963, it was only
recently that Schuhmacher (2005a) obtained error bounds for the weak
convergence. Schuhmacher considered dependent supposition, truncated the
individual point processes to 0--1 point processes and then applied Stein's
method to the latter. In this paper we take a different approach to the problem
by using Palm theory and Stein's method, thereby expressing the error bounds in
terms of the mean measures of the individual point processes, which is not
possible by Schuhmacher's approach. We consider locally dependent supposition
as a generalization of the locally dependent point process introduced in Chen
and Xia (2004) and apply the main theorem to the superposition of thinned point
processes and of renewal processes.
http://arxiv.org/abs/0901.2445
Author(s): Eric Cator and Leandro P.R. Pimentel
Abstract: The interplay between two-dimensional percolation growth models and
one-dimensional particle processes has always been a fruitful source of
interesting mathematical phenomena. In this paper we develop a connection
between the construction of Busemann functions in the Hammersley last-passage
percolation model with i.i.d. random weights, and the existence, ergodicity and
uniqueness of equilibrium measures for the related (multi-class) interacting
particle process. As we shall see, in the classical Hammersley model where each
point has weight one, this approach brings a new and rather geometrical
solution of the longest increasing subsequence problem, as well as a detailed
description of the scaling behavior of the Busemann function along different
directions.
http://arxiv.org/abs/0901.2450
Author(s): J. G. Dai and Wuqin Lin
Abstract: We consider a class of stochastic processing networks. Assume that the
networks satisfy a complete resource pooling condition. We prove that each
maximum pressure policy asymptotically minimizes the workload process in a
stochastic processing network in heavy traffic. We also show that, under each
quadratic holding cost structure, there is a maximum pressure policy that
asymptotically minimizes the holding cost. A key to the optimality proofs is to
prove a state space collapse result and a heavy traffic limit theorem for the
network processes under a maximum pressure policy. We extend a framework of
Bramson [Queueing Systems Theory Appl. 30 (1998) 89--148] and Williams
[Queueing Systems Theory Appl. 30 (1998b) 5--25] from the multiclass queueing
network setting to the stochastic processing network setting to prove the state
space collapse result and the heavy traffic limit theorem. The extension can be
adapted to other studies of stochastic processing networks.
http://arxiv.org/abs/0901.2451
Author(s): Stephen B. Connor and Gersende Fort
Abstract: We consider a form of state-dependent drift condition for a general Markov
chain, whereby the chain subsampled at some deterministic time satisfies a
geometric Foster-Lyapunov condition. We present sufficient criteria for such a
drift condition to exist, and use these to partially answer a question posed by
Connor & Kendall (2007) concerning the existence of so-called 'tame' Markov
chains. Furthermore, we show that our 'subsampled drift condition' implies the
existence of finite moments for the return time to a small set.
http://arxiv.org/abs/0901.2453
Author(s): Ester Gabetta and Eugenio Regazzini
Abstract: We prove that the solution of the Kac analogue of Boltzmann's equation can be
viewed as a probability distribution of a sum of a random number of random
variables. This fact allows us to study convergence to equilibrium by means of
a few classical statements pertaining to the central limit theorem. In
particular, a new proof of the convergence to the Maxwellian distribution is
provided, with a rate information both under the sole hypothesis that the
initial energy is finite and under the additional condition that the initial
distribution has finite moment of order $2+\delta$ for some $\delta$ in
$(0,1]$. Moreover, it is proved that finiteness of initial energy is necessary
in order that the solution of Kac's equation can converge weakly. While this
statement may seem to be intuitively clear, to our knowledge there is no proof
of it as yet.
http://arxiv.org/abs/0901.2464
Author(s): Wei-Dong Liu and Zhengyan Lin and Qi-Man Shao
Abstract: Let $\mathbf{X}_1,...,\mathbf{X}_n$ be a random sample from a $p$-dimensional
population distribution. Assume that $c_1n^{\alpha}\leq p\leq c_2n^{\alpha}$
for some positive constants $c_1,c_2$ and $\alpha$. In this paper we introduce
a new statistic for testing independence of the $p$-variates of the population
and prove that the limiting distribution is the extreme distribution of type I
with a rate of convergence $O((\log n)^{5/2}/\sqrt{n})$. This is much faster
than $O(1/\log n)$, a typical convergence rate for this type of extreme
distribution. A simulation study and application to stochastic optimization are
discussed.
http://arxiv.org/abs/0901.2468
Author(s): Amarjit Budhiraja and Kevin Ross
Abstract: A singular stochastic control problem with state constraints in
two-dimensions is studied. We show that the value function is $C^1$ and its
directional derivatives are the value functions of certain optimal stopping
problems. Guided by the optimal stopping problem, we then introduce the
associated no-action region and the free boundary and show that, under
appropriate conditions, an optimally controlled process is a Brownian motion in
the no-action region with reflection at the free boundary. This proves a
conjecture of Martins, Shreve and Soner [SIAM J. Control Optim. 34 (1996)
2133--2171] on the form of an optimal control for this class of singular
control problems. An important issue in our analysis is that the running cost
is Lipschitz but not $C^1$. This lack of smoothness is one of the key obstacles
in establishing regularity of the free boundary and of the value function. We
show that the free boundary is Lipschitz and that the value function is $C^2$
in the interior of the no-action region. We then use a verification argument
applied to a suitable $C^2$ approximation of the value function to establish
optimality of the conjectured control.
http://arxiv.org/abs/0901.2474
Author(s): Daniel Remenik
Abstract: We study a contact process running in a random environment in $\mathbb {Z}^d$
where sites flip, independently of each other, between blocking and nonblocking
states, and the contact process is restricted to live in the space given by
nonblocked sites. We give a partial description of the phase diagram of the
process, showing in particular that, depending on the flip rates of the
environment, survival of the contact process may or may not be possible for
large values of the birth rate. We prove block conditions for the process that
parallel the ones for the ordinary contact process and use these to conclude
that the critical process dies out and that the complete convergence theorem
holds in the supercritical case.
http://arxiv.org/abs/0901.2480
Author(s): Ingrid Carbone and Aljosa Volcic (University of Calabria - Italy)
Abstract: In this paper we consider permutations of sequences of partitions, obtaining
a result which parallels von Neumann's theorem on permutations of dense
sequences and uniformly distributed sequences of points.
http://arxiv.org/abs/0901.2531
Author(s): J. Harnad and A.Yu. Orlov
Abstract: A new representation of the 2N fold integrals appearing in various two-matrix
models that admit reductions to integrals over their eigenvalues is given in
terms of vacuum state expectation values of operator products formed from
two-component free fermions. This is used to derive the perturbation series for
these integrals under deformations induced by exponential weight factors in the
measure, expressed as double and quadruple Schur function expansions,
generalizing results obtained earlier for certain two-matrix models. Links with
the coupled two-component KP hierarchy and the two-component Toda lattice
hierarchy are also derived.
http://arxiv.org/abs/math-ph/0512056
Author(s): Xianming Liu and Jinqiao Duan and Jicheng Liu and Peter E. Kloeden
Abstract: Dynamical systems driven by Gaussian noises have been considered extensively
in modeling, simulation and theory. However, complex systems in engineering and
science are often subject to non-Gaussian fluctuations or uncertainties. A
coupled dynamical system under non- Gaussian Levy noises is considered. After
discussing cocycle prop- erty, stationary orbits and random attractors, a
synchronization phe- nomenon is shown to occur, when the drift terms of the
coupled system satisfy certain dissipativity and integrability conditions. The
synchro- nization result implies that coupled dynamical systems share a dy-
namical feature in some asymptotic sense.
http://arxiv.org/abs/0901.2446
Author(s): E. Ostrovsky
Abstract: We obtain in this paper using the saddle point method the expression for the
exact asymptotic for the tail of maximum of smooth (twice continuous
differentiable) random field (process) distribution.
http://arxiv.org/abs/0901.2714
Author(s): Dmitry Dolgopyat and Leonid Koralov
Abstract: We consider a process on $\mathbb{T}^2$, which consists of fast motion along
the stream lines of an incompressible periodic vector field perturbed by white
noise. It gives rise to a process on the graph naturally associated to the
structure of the stream lines of the unperturbed flow. It has been shown by
Freidlin and Wentzell [Random Perturbations of Dynamical Systems, 2nd ed.
Springer, New York (1998)] and [Mem. Amer. Math. Soc. 109 (1994)] that if the
stream function of the flow is periodic, then the corresponding process on the
graph weakly converges to a Markov process. We consider the situation where the
stream function is not periodic, and the flow (when considered on the torus)
has an ergodic component of positive measure. We show that if the rotation
number is Diophantine, then the process on the graph still converges to a
Markov process, which spends a positive proportion of time in the vertex
corresponding to the ergodic component of the flow.
http://arxiv.org/abs/0901.2776
Author(s): Heikki Sepp\"al\"a
Abstract: Given an increasing function $H:[0,1)\to [0,\infty)$ and $$
A_n(H):=\inf_{\tau\in \mathcal{T}_n}(\sum_{i=1}^n \int_{t_{i-1}}^{t_i}
(t_i-t)H^2(t)dt)^{{1/2}}, $$ where $\mathcal{T}_n:=\{\tau=(t_i)_{i=0}^n:
0=t_0
http://arxiv.org/abs/0901.2777
Author(s): Jin Ma and Jianfeng Zhang and Ziyu Zheng
Abstract: In this paper, we propose a new notion of Forward--Backward Martingale
Problem (FBMP), and study its relationship with the weak solution to the
forward--backward stochastic differential equations (FBSDEs). The FBMP extends
the idea of the well-known (forward) martingale problem of Stroock and
Varadhan, but it is structured specifically to fit the nature of an FBSDE. We
first prove a general sufficient condition for the existence of the solution to
the FBMP. In the Markovian case with uniformly continuous coefficients, we show
that the weak solution to the FBSDE (or equivalently, the solution to the FBMP)
does exist. Moreover, we prove that the uniqueness of the FBMP (whence the
uniqueness of the weak solution) is determined by the uniqueness of the
viscosity solution of the corresponding quasilinear PDE.
http://arxiv.org/abs/0901.2790
Author(s): Olav Kallenberg
Abstract: Let $\xi$ be a Dawson--Watanabe superprocess in $\mathbb{R}^d$ such that
$\xi_t$ is a.s. locally finite for every $t\geq 0$. Then for $d\geq2$ and fixed
$t>0$, the singular random measure $\xi_t$ can be a.s. approximated by suitably
normalized restrictions of Lebesgue measure to the $\varepsilon$-neighborhoods
of $\operatorname {supp}\xi_t$. When $d\geq3$, the local distributions of
$\xi_t$ near a hitting point can be approximated in total variation by those of
a stationary and self-similar pseudo-random measure $\tilde{\xi}$. By contrast,
the corresponding distributions for $d=2$ are locally invariant. Further
results include improvements of some classical extinction criteria and some
limiting properties of hitting probabilities. Our main proofs are based on a
detailed analysis of the historical structure of $\xi$.
http://arxiv.org/abs/0901.2840
Author(s): Patrizia Berti and Luca Pratelli and Pietro Rigo
Abstract: Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{N}$ the
class of those $F\in\mathcal{F}$ satisfying $P(F)\in\{0,1\}$. For each
$\mathcal{G}\subset\mathcal{F}$, define
$\overline{\mathcal{G}}=\sigma(\mathcal{G}\cup\mathcal {N})$. Necessary and
sufficient conditions for
$\overline{\mathcal{A}}\cap\overline{\mathcal{B}}=\overline {\mathcal
{A}\cap\mathcal{B}}$, where $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ are
sub-$\sigma$-fields, are given. These conditions are then applied to the
(two-component) Gibbs sampler. Suppose $X$ and $Y$ are the coordinate
projections on $(\Omega,\mathcal{F})=(\mathcal{X}\times\mathcal{Y},\mathcal
{U}\otimes \mathcal{V})$ where $(\mathcal{X},\mathcal{U})$ and
$(\mathcal{Y},\mathcal{V})$ are measurable spaces. Let $(X_n,Y_n)_{n\geq0}$ be
the Gibbs chain for $P$. Then, the SLLN holds for $(X_n,Y_n)$ if and only if
$\overline{\sigma(X)}\cap\overline{\sigma(Y)}=\mathcal{N}$, or equivalently if
and only if $P(X\in U)P(Y\in V)=0$ whenever $U\in\mathcal{U}$,
$V\in\mathcal{V}$ and $P(U\times V)=P(U^c\times V^c)=0$. The latter condition
is also equivalent to ergodicity of $(X_n,Y_n)$, on a certain subset
$S_0\subset\Omega$, in case $\mathcal{F}=\mathcal{U}\otimes\mathcal{V}$ is
countably generated and $P$ absolutely continuous with respect to a product
measure.
http://arxiv.org/abs/0901.2851
Author(s): F. Kn\"able
Abstract: We solve a time-dependent linear SPDE with additive Levy noise in the mild
and weak sense. Existence of a generalized invariant measure for the associated
transition semigroup is established and the generator is characterized on the
corresponding L^2-space. The square field operator is calculated, allowing to
derive a Poincare and a Harnack inequality.
http://arxiv.org/abs/0901.2887
Author(s): Robert Wooster
Abstract: We examine the question of existence and uniqueness of evolution systems of
measures for non-autonomous Ornstein-Uhlenbeck-type processes with jumps. In
particular, we give examples where we explicitly compute the densities of such
families of measures.
http://arxiv.org/abs/0901.2899
Author(s): Francesco Caravenna and Nicolas P\'etr\'elis
Abstract: In this paper we consider a model which describes a polymer chain interacting
with an infinity of equi-spaced linear interfaces. The distance between two
consecutive interfaces is denoted by T = T_N and is allowed to grow with the
size N of the polymer. When the polymer receives a positive reward for touching
the interfaces, its asymptotic behavior has been derived in a previous paper,
showing that a transition occurs when T_N \approx log(N). In the present paper,
we deal with the so-called depinning case, i.e., the polymer is repelled rather
than attracted by the interfaces. Using techniques from renewal theory, we
determine the scaling behavior of the model for large N as a function of T_N,
showing that two transitions occur, when T_N \approx N^{1/3} and when T_N
\approx N^{1/2} respectively.
http://arxiv.org/abs/0901.2902
Author(s): Kjetil Roysland
Abstract: Marginal structural models were introduced in order to provide estimates of
causal effects from interventions based on observational studies in
epidemiological research. We present a variant of the marginal structural
strategy in continuous time using martingale theory and marked point processes.
This offers a mathematical interpretation of marginal structural models that
has not been available before.
Our approach starts with a characterization of reasonable models of
randomized trials in terms of local independence. Such a model gives a
martingale measure that is equivalent to the observational measure. The
continuous time likelihood ratio process with respect to these two probability
measures corresponds to the weights in a discrete time marginal structural
model. In order to do inference for the new measure, we can simulate sampling
using the observed data weighted by this likelihood ratio.
http://arxiv.org/abs/0901.2593
Author(s): L. de Francesco Albasini and N. Sabadini and R.F.C. Walters
Abstract: We describe an algebra for composing automata in which the actions have
probabilities. We illustrate by showing how to calculate the probability of
reaching deadlock in k steps in a model of the classical Dining Philosopher
problem, and show, using the Perron-Frobenius Theorem, that this probability
tends to 1 as k tends to infinity.
http://arxiv.org/abs/0901.2434
Author(s): John H. Elton and Theodore P. Hill
Abstract: The conclusion of the classical ham sandwich theorem of Banach and Steinhaus
may be strengthened: there always exists a common bisecting hyperplane that
touches each of the sets, that is, intersects the closure of each set. Hence,
if the knife is smeared with mayonnaise, a cut can always be made so that it
will not only simultaneously bisect each of the ingredients, but it will also
spread mayonnaise on each. A discrete analog of this theorem says that n finite
nonempty sets in n-dimensional Euclidean space can always be simultaneously
bisected by a single hyperplane that contains at least one point in each set.
More generally, for n compactly-supported positive finite Borel measures in
Euclidean n-space, there is always a hyperplane that bisects each of the
measures and intersects the support of each measure.
http://arxiv.org/abs/0901.2589
Author(s): Achim Klenke and Mario Oeler
Abstract: Dawson and Perkins (1998) constructed a stochastic model of an interacting
two-type population indexed by a countable site space which locally undergoes a
mutually catalytic branching mechanism. Klenke and Mytnik (2009) showed that as
the branching rate approaches infinity the process converges to a process that
is called the infinite rate mutually catalytic branching process. It is most
conveniently characterised as the solution to a certain martingale problem.
While Klenke and Mytnik used a noise equation approach in order to construct a
solution to this martingale problem, the aim of this paper is to provide a
Trotter type construction.
http://arxiv.org/abs/0901.2993
Author(s): A. Gaudilliere
Abstract: These notes constitute the introduction to potential theory I exposed at the
XIIth brazilian school of probability inside Elisabetta Scoppola's Introduction
to Metastability.
http://arxiv.org/abs/0901.3053
Author(s): Nicolas Fournier
Abstract: We consider the problem of the simulation of Levy-driven stochastic
differential equations. It is generally impossible to simulate the increments
of a Levy-process. Thus in addition to an Euler scheme, we have to simulate
approximately these increments. We use a method in which the large jumps are
simulated exactly, while the small jumps are approximated by Gaussian
variables. Using some recent results of Rio about the central limit theorem, in
the spirit of the famous paper by Komlos-Major-Tsunady, we derive an estimate
for the strong error of this numerical scheme. This error remains reasonnable
when the Levy measure is very singular near 0, which is not the case when
neglecting the small jumps.
In the same spirit, we study the problem of the approximation of a
Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large
jumps.
http://arxiv.org/abs/0901.3082
Author(s): Jan Mandel and Loren Cobb and and Jonathan D. Beezley
Abstract: Convergence of the ensemble Kalman filter in the limit for large ensembles to
the Kalman filter is proved. In each step of the filter, convergence of the
ensemble sample covariance follows from a weak law of large numbers for
exchangeable random variables, Slutsky's theorem gives weak convergence of
ensemble members, and $L^p$ bounds on the ensemble then give $L^p$ convergence.
http://arxiv.org/abs/0901.2951
Author(s): Antoine Ayache (LPP) and Pierre R. Bertrand (INRIA Saclay - Ile de France)
Abstract: In Ayache and Taqqu (2005), the multifractional Brownian (mBm) motion is
obtained by replacing the constant parameter $H$ of the fractional Brownian
motion (fBm) by a smooth enough functional parameter $H(.)$ depending on the
time $t$. Here, we consider the process $Z$ obtained by replacing in the
wavelet expansion of the fBm the index $H$ by a function $H(.)$ depending on
the dyadic point $k/2^j$. This process was introduced in Benassi et al (2000)
to model fBm with piece-wise constant Hurst index and continuous paths. In this
work, we investigate the case where the functional parameter satisfies an
uniform H\"older condition of order $\beta>\sup_{t\in \rit} H(t)$ and ones
shows that, in this case, the process $Z$ is very similar to the mBm in the
following senses: i) the difference between $Z$ and a mBm satisfies an uniform
H\"older condition of order $d>\sup_{t\in \R} H(t)$; ii) as a by product, one
deduces that at each point $t\in \R$ the pointwise H\"older exponent of $Z$ is
$H(t)$ and that $Z$ is tangent to a fBm with Hurst parameter $H(t)$.
http://arxiv.org/abs/0901.2808
Author(s): Wendell H. Fleming and Hidehiro Kaise and Shuenn-Jyi Sheu
Abstract: In the Maslov idempotent probability calculus, expectations of random
variables are defined so as to be linear with respect to max-plus addition and
scalar multiplication. This paper considers control problems in which the
objective is to minimize the max-plus expectation of some max-plus additive
running cost. Such problems arise naturally as limits of some types of risk
sensitive stochastic control problems. The value function is a viscosity
solution to a quasivariational inequality (QVI) of dynamic programming.
Equivalence of this QVI to a nonlinear parabolic PDE with discontinuous
Hamiltonian is used to prove a comparison theorem for viscosity sub- and
super-solutions. An example from math finance is given, and an application in
nonlinear H-infinity control is sketched.
http://arxiv.org/abs/0901.3007
Author(s): M. F. Bayramoglu and A. \"Ozg\"ur Y{\i}lmaz
Abstract: We show that any joint probability mass function (PMF) can be expressed as a
product of parity check factors and factors of degree one, if the alphabet size
is appropriate for defining a parity check equation. In other words,
marginalization or maximization of a joint PMF is equivalent to a decoding task
as long as a finite field can be constructed over the alphabet of the PMF. In
factor graph terminology this claim means that a factor graph representing such
a joint PMF always has an equivalent Tanner graph. We provide a systematic
method based on the Hilbert space of PMFs and orthogonal projections for
obtaining this factorization.
http://arxiv.org/abs/0901.3056
Author(s): Hanna D\"oring and Peter Eichelsbacher
Abstract: We prove a moderate deviation principle for subgraph count statistics of
Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation
principle for the trace of a power of a Bernoulli random matrix. It is done via
an estimation of the log-Laplace transform and the Gaertner-Ellis theorem. We
obtain upper bounds on the upper tail probabilities of the number of
occurrences of small subgraphs. The method of proof is used to show
supplemental moderate deviation principles for a class of symmetric statistics,
including non-degenerate U-statistics with independent or Markovian entries.
http://arxiv.org/abs/0901.3246
Author(s): Enrico Valdinoci
Abstract: This note illustrates how a simple random walk with possibly long jumps is
related to fractional powers of the Laplace operator. The exposition is
elementary and self-contained.
http://arxiv.org/abs/0901.3261
Author(s): Mathew D. Penrose and Andrew R. Wade
Abstract: Suppose that under the action of gravity, liquid drains through the unit
$d$-cube via a minimal-length network of channels constrained to pass through
random sites and to flow with nonnegative component in one of the canonical
orthogonal basis directions of $\R^d$, $d \geq 2$. The resulting network is a
version of the so-called minimal directed spanning tree. We give laws of large
numbers and convergence in distribution results on the large-sample asymptotic
behaviour of the total power-weighted edge-length of the network on uniform
random points in $(0,1)^d$. The distributional results exhibit a
weight-dependent phase transition between Gaussian and boundary-effect-derived
distributions. These boundary contributions are characterized in terms of
limits of the so-called on-line nearest-neighbour graph, a natural model of
spatial network evolution, for which we also present some new results. Also, we
give a convergence in distribution result for the length of the longest edge in
the drainage network; when $d=2$, the limit is expressed in terms of
Dickman-type variables.
http://arxiv.org/abs/0901.3297
Author(s): F.Michel Dekking and Karoly Simon and and Balazs Szekely
Abstract: In this paper we consider a family of random Cantor sets on the line and
consider the question whether the condition that the sum of the Hausdorff
dimensions is larger than one implies the existence of interior points in the
difference set of two independent copies. We give a new and complete proof that
this is the case for the random Cantor sets introduced by Per Larsson.
http://arxiv.org/abs/0901.3304
Author(s): Aijun Du and Jinqiao Duan
Abstract: Complex systems display variability over a broad range of spatial and
temporal scales. Some scales are unresolved due to computational limitations.
The impact of these unresolved scales on the resolved scales needs to be
parameterized or taken into account. One stochastic parameterization scheme is
devised to take the effects of unresolved scales into account, in the context
of solving a nonlinear partial differential equation with memory (a
time-integral term), via large eddy simulations.
The obtained large eddy simulation model is a stochastic partial differential
equation. Numerical experiments are performed to compare the solutions of the
original system and of the stochastic large eddy simulation model.
http://arxiv.org/abs/0901.3312
Author(s): K\'aroly J. B\"or\"oczky and Rolf Schneider
Abstract: For a given convex body K in $R^d$, a random polytope $K^{(n)}$ is defined
(essentially) as the intersection of $n$ independent closed halfspaces
containing $K$ and having an isotropic and (in a specified sense) uniform
distribution. We prove upper and lower bounds, of optimal orders, for the
difference of the mean widths of $K^{(n)}$ and K, as n tends to infinity. For a
simplicial polytope P, a precise asymptotic formula for the difference of the
mean widths of $P^{(n)}$ and P is obtained.
http://arxiv.org/abs/0901.3343
Author(s): Michael Bj\"orklund
Abstract: We generalize the asymptotic shape theorem in first passage percolation on
$\Z^d$ to cover the case of general semimetrics. We prove a structure theorem
for equivariant semimetrics on topological groups and an extended version of
the maximal inequality for $\Z^d$-cocycles by D. Boivin and Y. Derriennic in
the vector-valued case. This inequality will imply a very general form of
Kingman's subadditive ergodic theorem. For certain classes of generalized first
passage percolation we prove further structure theorems and provide rates of
convergence in the asymptotic shape theorem. We also establish a general form
of the multiplicative ergodic theorem by A. Karlsson and F. Ledrappier for
cocycles with values in separable Banach spaces with the Radon-Nikod\'ym
property.
http://arxiv.org/abs/0901.3449
Author(s): Emmanuel Jacob (PMA)
Abstract: The integrated Brownian motion is sometimes known as the Langevin process.
Lachal studied several excursion laws induced by the latter. Here we follow a
different point of view developed by Pitman for general stationary processes.
We first construct a stationary Langevin process and then determine explicitly
its stationary excursion measure. This is then used to provide new descriptions
of It\^o's excursion measure of the Langevin process reflected at a completely
inelastic boundary, which has been introduced recently by Bertoin.
http://arxiv.org/abs/0901.3464
Author(s): Sylvain Rubenthaler (JAD)
Abstract: The Bird and Nanbu systems are particle systems used to approximate the
solution of Boltzmann mollified equation. In particular, they have the
propagation of chaos property. Following [GM94], we use coupling techniques and
resultson branching processes to write an expansion of the error in the
propagation of chaos in terms of the number of particles, for slightly more
general systems than the ones cited above. As explained in [DMPR] and [DMPR09],
this result will lead to the proof of the convergence of U-statistics for these
systems.
http://arxiv.org/abs/0901.3476
Author(s): Mathew D. Penrose
Abstract: Consider throwing $n$ balls at random into $m$ urns, each ball landing in urn
$i$ with probability $p_i$. Let $S$ be the resulting number of singletons,
i.e., urns containing just one ball. We give an error bound for the Kolmogorov
distance from $S$ to the normal, and estimates on its variance. These show that
if $n$, $m$ and $(p_i, 1 \leq i \leq m)$ vary in such a way that $\sup_i p_i =
O(n^{-1})$, then $S$ satisfies a CLT if and only if $n^2 \sum_i p_i^2$ tends to
infinity, and demonstrate an optimal rate of convergence in the CLT in this
case. In the uniform case $(p_i \equiv m^{-1}) with $m$ and $n$ growing
proportionately, we provide bounds with better asymptotic constants. The proof
of the error bounds are based on Stein's method via size-biased couplings.
http://arxiv.org/abs/0901.3493
Author(s): Fei Li and Yifeng Xue
Abstract: We define zonal polynomials of quaternion matrix argument and deduce some
important formulae of zonal polynomials and hypergeometric functions of
quaternion matrix argument. As an application, we give the distributions of the
largest and smallest eigenvalues of a quaternion central Wishart matrix
$W\sim\mathbb{Q}W(n,\Sigma)$, respectively.
http://arxiv.org/abs/0901.3379
Author(s): K\'aroly J. B\"or\"oczky and Ferenc Fodor and Daniel Hug
Abstract: Let K be a d-dimensional convex body, and let $K^{(n)}$ be the intersection
of n halfspaces containing $K$ whose bounding hyperplanes are independent and
identically distributed. Under suitable distributional assumptions, we prove an
asymptotic formula for the expectation of the difference of the mean widths of
$K^{(n)}$ and K, and another asymptotic formula for the expectation of the
number of facets of $K^{(n)}$. These results are achieved by establishing an
asymptotic result on weighted volume approximation of $K$ and by "dualizing" it
using polarity.
http://arxiv.org/abs/0901.3419
Author(s): S.C. Lim and L.P. Teo
Abstract: This paper considers a generalization of Gaussian random field with
covariance function of Whittle-Mat$\acute{\text{e}}$rn family. Such a random
field can be obtained as the solution to the fractional stochastic differential
equation with two fractional orders. Asymptotic properties of the covariance
functions belonging to this generalized Whittle-Mat$\acute{\text{e}}$rn family
are studied, which are used to deduce the sample path properties of the random
field. The Whittle-Mat$\acute{\text{e}}$rn field has been widely used in
modeling geostatistical data such as sea beam data, wind speed, field
temperature and soil data. In this article we show that generalized
Whittle-Mat$\acute{\text{e}}$rn field provides a more flexible model for wind
speed data.
http://arxiv.org/abs/0901.3581
Author(s): Alan Frieze and Santosh Vempala and Juan Vera
Abstract: We propose the following model of a random graph on n vertices. Let F be a
distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le
i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained
by picking a random point X from F and defining a graph on n vertices whose
edges are pairs ij for which X_{ij} \le p. The standard Erd\H{o}s-R\'{e}nyi
model is the special case when F is uniform on the 0-1 unit cube. We examine
basic properties such as the connectivity threshold for quite general
distributions. We also consider cases where the X_{ij} are the edge weights in
some random instance of a combinatorial optimization problem. By choosing
suitable distributions, we can capture random graphs with interesting
properties such as triangle-free random graphs and weighted random graphs with
bounded total weight.
http://arxiv.org/abs/0901.3697
Author(s): Krzysztof Szajowski
Abstract: We register a random sequence constructed based on Markov processes by
switching between them. At two random moments $\theta_1$, $\theta_2$, where
$0\leq \theta_1 \leq \theta_2$, the source of observations is changed. In
effect the number of homogeneous segments is random. The transition
probabilities of each process are known and \emph{a priori} distribution of the
disorder moments is given. The various questions are formulated concerning the
distribution changes in the model in the former research. The random number of
distributional segments creates new problems in solutions of the problems
formulated for model with deterministic number of segments. Two cases are
presented in details. In the first one the objectives is to stop on between the
disorder moments and in the second one our objective is to find the strategy
which immediately detects the distribution changes. Both problems are
reformulated to optimal stopping of the observed sequences. The detailed
analysis of the problem is presented to show the form of optimal decision
function.
http://arxiv.org/abs/0901.3795
Author(s): Mohammud Foondun and Davar Khoshnevisan
Abstract: Consider a stochastic heat equation $\partial_t u = \kappa
\partial^2_{xx}u+\sigma(u)\dot{w}$ for a space-time white noise $\dot{w}$ and a
constant $\kappa>0$. Under some suitable conditions on the the initial function
$u_0$ and $\sigma$, we show that the quantity
\limsup_{t\to\infty}t^{-1}\ln\E(\sup_{x\in\R} |u_t(x)|^2) is bounded away from
zero and infinity by explicit multiples of $1/\kappa$. Our proof works by
demonstrating quantitatively that the peaks of the stochastic process $x\mapsto
u_t(x)$ are highly concentrated for infinitely-many large values of $t$. In the
special case of the parabolic Anderson model--where $\sigma(u)= \lambda u$ for
some $\lambda>0$--this "peaking" is a way to make precise the notion of
physical intermittency.
http://arxiv.org/abs/0901.3814
Author(s): Carl Mueller and Roger Tribe
Abstract: In this paper a stochastic reaction diffusion system is considered, which
models the spread of a finite population reacting with a non-renewable resource
in the presence of individual based noise. A two-parameter phase diagram is
established to describe the large time evolution, distinguishing between
certain death or possible life of the population.
http://arxiv.org/abs/0901.3859
Author(s): Takahiro Aoyama and Alexander Lindner and Makoto Maejima
Abstract: Recently, many classes of infinitely divisible distributions on R^d have been
characterized in several ways. Among others, the first way is to use Levy
measures, the second one is to use transformations of Levy measures, and the
third one is to use mappings of infinitely divisible distributions defined by
stochastic integrals with respect to Levy processes. In this paper, we are
concerned with a class of mappings, by which we construct new classes of
infinitely divisible distributions on R^d. Then we study a special case in R^1,
which is the class of infinitely divisible distributions without Gaussian parts
generated by stochastic integrals with respect to a fixed compound Poisson
processes on R^1. This is closely related to the Goldie-Steutel-Bondesson
class.
http://arxiv.org/abs/0901.3874
Author(s): Damir Filipovic and Eberhard Mayerhofer
Abstract: We revisit affine diffusion processes on general and on the canonical state
space in particular. A detailed study of theoretic and applied aspects of this
class of Markov processes is given. In particular, we derive admissibility
conditions and provide a full proof of existence and uniqueness through
stochastic invariance of the canonical state space. Existence of exponential
moments and the full range of validity of the affine transform formula are
established. This is applied to the pricing of bond and stock options, which is
illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models.
http://arxiv.org/abs/0901.4003
Author(s): Hiroyuki Matsumoto
Abstract: By adopting the upper half space realizations of the real, complex and
quaternionic hyperbolic spaces and solving the corresponding stochastic
differential equations, we can represent the Brownian motions on these
classical families of the hyperbolic spaces as explicit Wiener functionals.
Using the representations, we show that the almost sure convergence of the
Brownian motions and the central limit theorems for the radial components as
time tends to infinity are easily obtained. We also give a straightforward
strategy to obtain the explicit expressions for the Poisson kernels by
combining the representations with some results on the distributions of the
random variables which are defined by the perpetual (infinite) integrals of the
usual geometric Brownian motions with negative drifts.
http://arxiv.org/abs/0901.4028
Author(s): Anne Fey and Lionel Levine and Yuval Peres
Abstract: We study the abelian sandpile growth model, where n particles are added at
the origin on a stable background configuration in Z^d. Any site with at least
2d particles then topples by sending one particle to each neighbor. We find
that with constant background height h <= 2d-2, the diameter of the set of
sites that topple has order n^{1/d}. This was previously known only for h
http://arxiv.org/abs/0901.3805
Author(s): A.V. Bobylev and C. Cercignani and I.M. Gamba
Abstract: We consider generalizations of kinetic granular gas models given by Boltzmann
equations of Maxwell type. These type of models for non-linear elastic or
inelastic interactions, have many applications in physics, dynamics of granular
gases, economy, etc. We present the problem and develop its form in the space
of characteristic functions, i.e. Fourier transforms of probability measures,
from a very general point of view, including those with arbitrary polynomial
non-linearities and in any dimension space. We find a whole class of
generalized Maxwell models that satisfy properties that characterize the
existence and asymptotic of dynamically scaled or self-similar solutions, often
referred as {\em homogeneous cooling states}. Of particular interest is a
concept interpreted as an operator generalization of usual Lipschitz conditions
which allows to describe the behavior of solutions to the corresponding initial
value problem. In particular, we present, in the most general case, existence
of self similar solutions and study, in the sense of probability measures, the
convergence of dynamically scaled solutions associated with the Cauchy problem
to those self-similar solutions, as time goes to infinity. In addition we show
that the properties of these self-similar solutions lead to non classical
equilibrium stable states exhibiting power tails. These results apply to
different specific problems related to the Boltzmann equation (with elastic and
inelastic interactions) and show that all physically relevant properties of
solutions follow directly from the general theory developed in this
presentation.
http://arxiv.org/abs/0901.3864
Author(s): Noga Alon and Ori Gurel-Gurevich and Eyal Lubetzky
Abstract: In the classical balls-and-bins paradigm, where n balls are placed
independently and uniformly in n bins, typically the number of bins with at
least two balls in them has order n and the maximum number of balls in a bin
has order (log n)/(log log n). It is well known that when each round offers k
independent uniform options for bins, it is possible to typically achieve a
constant maximal load if and only if k is at least of order (log n). Moreover,
it is possible whp to avoid any collisions between (n/2) balls if (k> log_2 n).
In this work, we extend this into the setting where only m bits of memory are
available. We establish a tradeoff between the number of choices k and the
memory m, dictated by the quantity km/n. Roughly put, we show that for (k m)
larger than n, one can achieve a constant maximal load, while for (k m) smaller
than n no substantial improvement can be gained over the case k=1 (i.e., a
random allocation).
For any (k = \Omega(log n)) and (m = \Omega(log^2 n)), one can achieve a
constant load whp if (k m = \Omega(n)), yet the load is unbounded if (k m
=o(n)). Similarly, if (k m > C n) then (n/2) balls can be allocated without any
collisions whp, whereas for (k m < \epsilon n) there are typically order n
collisions. Furthermore, we show that the load is whp at least log(n/m)/[log k
+ log log(n/m)]. In particular, for k=polylog(n), if m = n^{1-\delta} the
optimal maximal load is of order (log n)/(log log n) (the same as in the case
k=1), while m=2n suffices to ensure a constant load. Finally, we analyze
non-adaptive allocation algorithms and give tight upper and lower bounds for
their performance.
http://arxiv.org/abs/0901.4056
Author(s): Mohammud Foondun
Abstract: We consider the Dirichlet form given by
\sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial
f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times
\bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption that the $\{a_{ij}\}$ are
symmetric and uniformly elliptic and with suitable conditions on $J$, the
nonlocal part, we obtain upper and lower bounds on the heat kernel of the
Dirichlet form. We also prove a Harnack inequality and a regularity theorem for
functions that are harmonic with respect to $\sE$.
http://arxiv.org/abs/0901.4127
Author(s): Christian Houdr\'e and Hua Xu
Abstract: The limiting shape of the random Young tableaux associated to the
inhomogeneous word problem is identified as a multidimensional Brownian
functional. This functional is thus identical in law to the spectrum of a
certain matrix ensemble. The Poissonized word problem is also studied, and the
asymptotic behavior of the shape analyzed.
http://arxiv.org/abs/0901.4138
Author(s): Jian Ding and Eyal Lubetzky and Yuval Peres
Abstract: In the heat-bath Glauber dynamics for the Ising model on the lattice,
physicists believe that the spectral gap of the continuous-time chain exhibits
the following behavior. For some critical inverse-temperature $\beta_c$, the
inverse-gap is bounded for $\beta < \beta_c$, polynomial in the surface area
for $\beta = \beta_c$ and exponential in it for $\beta > \beta_c$. This has
been proved for $\Z^2$ except at criticality. So far, the only underlying
geometry where the critical behavior has been confirmed is the complete graph.
Recently, the dynamics for the Ising model on a regular tree, also known as the
Bethe lattice, has been intensively studied. The facts that the inverse-gap is
bounded for $\beta < \beta_c$ and exponential for $\beta > \beta_c$ were
established, where $\beta_c$ is the critical spin-glass parameter, and the
tree-height $h$ plays the role of the surface area.
In this work, we complete the picture for the inverse-gap of the Ising model
on the $b$-ary tree, by showing that it is indeed polynomial in $h$ at
criticality. The degree of our polynomial bound does not depend on $b$, and
furthermore, this result holds under any boundary condition. We also obtain
analogous bounds for the mixing-time of the chain. In addition, we study the
near critical behavior, and show that for $\beta > \beta_c$, the inverse-gap
and mixing-time are both $\exp[\Theta((\beta-\beta_c) h)]$.
http://arxiv.org/abs/0901.4152
Author(s): Matti Lassas. Eero Saksman and Samuli Siltanen
Abstract: Bayesian solution of an inverse problem for indirect measurement $M = AU +
{\mathcal{E}}$ is considered, where $U$ is a function on a domain of $R^d$.
Here $A$ is a smoothing linear operator and $ {\mathcal{E}}$ is Gaussian white
noise. The data is a realization $m_k$ of the random variable $M_k = P_kA U+P_k
{\mathcal{E}}$, where $P_k$ is a linear, finite dimensional operator related to
measurement device. To allow computerized inversion, the unknown is discretized
as $U_n=T_nU$, where $T_n$ is a finite dimensional projection, leading to the
computational measurement model $M_{kn}=P_k A U_n + P_k {\mathcal{E}}$. Bayes
formula gives then the posterior distribution $\pi_{kn}(u_n |
m_{kn})\sim\pi_n(u_n) \exp(-{1/2}\|m_{kn} - P_kA u_n\|_2^2)$ in $R^d$, and the
mean $U^{CM}_{kn}:=\int u_n \pi_{kn}(u_n | m_k) du_n$ is considered as the
reconstruction of $U$. We discuss a systematic way of choosing prior
distributions $\prior_n$ for all $n\geq n_0>0$ by achieving them as projections
of a distribution in a infinite-dimensional limit case. Such choice of prior
distributions is {\em discretization-invariant} in the sense that $\prior_n$
represent the same {\em a priori} information for all $n$ and that the mean
$U^{CM}_{kn}$ converges to a limit estimate as $k,n\to\infty$. Gaussian
smoothness priors and wavelet-based Besov space priors are shown to be
discretization invariant. In particular, Bayesian inversion in dimension two
with $B^1_{11}$ prior is related to penalizing the $\ell^1$ norm of the wavelet
coefficients of $U$.
http://arxiv.org/abs/0901.4220
Author(s): Anders Bj\"orner
Abstract: A Markov chain is considered whose states are orderings of an underlying
fixed tree and whose transitions are local "random-to-front" reorderings,
driven by a probability distribution on subsets of the leaves. The eigenvalues
of the transition matrix are determined using Brown's theory of random walk on
semigroups.
http://arxiv.org/abs/0901.4278
Author(s): Mark Holmes
Abstract: We study a random walk that has a drift $\frac{\beta}{d}$ to the right when
located at a previously unvisited vertex and a drift $\frac{\mu}{d}$ to the
left otherwise. We prove that in high dimensions, for every $\mu$, the drift to
the right is a strictly increasing and continuous function of $\beta$, and that
there is precisely one value $\beta_0(\mu,d)$ for which the resulting speed is
zero.
http://arxiv.org/abs/0901.4393
Author(s): Peter Baxendale and Georgi Dimitroff
Abstract: We study some finite time transport properties of isotropic Brownian flows.
Under a certain nondegeneracy condition on the potential spectral measure, we
prove that uniform shrinking or expansion of balls under the flow over some
bounded time interval can happen with positive probability. We also provide a
control theorem for isotropic Brownian flows with drift. Finally, we apply the
above results to show that under the nondegeneracy condition the length of a
rectifiable curve evolving in an isotropic Brownian flow with strictly negative
top Lyapunov exponent converges to zero as $t\to \infty$ with positive
probability.
http://arxiv.org/abs/0901.4414
Author(s): Alexander V. Gnedin
Abstract: Theory of Kingman's partition structures has two culminating points: the
general paintbox representation, relating finite partitions to hypothetical
infinite populations via a natural sampling procedure, known as Kingman's
paintbox; a central example of the theory - the Ewens-Pitman two-parameter
family of partitions.
In these notes we further develop the theory by passing to structures
enriched by the order on the collection of categories; extending the class of
tractable models by exploring the idea of regeneration; analysing regenerative
properties of the Ewens-Pitman partitions; studying asymptotic features of the
regenerative compositions.
http://arxiv.org/abs/0901.4444
Author(s): Jean-Christophe Breton (MIA) and Ivan Nourdin (PMA) and Giovanni Peccati (MODAL'X)
Abstract: In this short note, we show how to use concentration inequalities in order to
build exact confidence intervals for the Hurst parameter associated with a
one-dimensional fractional Brownian motion
http://arxiv.org/abs/0901.4456
Author(s): Mark Adler and Nicolas Orantin and Pierre van Moerbeke
Abstract: Consider non-intersecting Brownian motions on the line leaving from the
origin and forced to two arbitrary points. Letting the number of Brownian
particles tend to infinity, and upon rescaling, there is a point of
bifurcation, where the support of the density of particles goes from one
interval to two intervals. In this paper, we show that at that very point of
bifurcation a cusp appears, near which the Brownian paths fluctuate like the
Pearcey process. This is a universality result within this class of problems.
Tracy and Widom obtained such a result in the symmetric case, when the two
target points are symmetric with regard to the origin. This asymmetry enabled
us to improve considerably a result concerning the non-linear partial
differential equations governing the transition probabilities for the Pearcey
process, obtained by Adler and van Moerbeke.
http://arxiv.org/abs/0901.4520
Author(s): Itai Benjamini and Asaf Nachmias and Yuval Peres
Abstract: We show that the critical probability for percolation on a d-regular
non-amenable graph of large girth is close to the critical probability for
percolation on an infinite d-regular tree. This is a special case of a
conjecture due to O. Schramm on the locality of p_c. We also prove a finite
analogue of the conjecture for expander graphs.
http://arxiv.org/abs/0901.4616
Author(s): Lo\"ic Herv\'e (IRMAR) and Fran\c{c}oise P\`ene (LM)
Abstract: Nagaev's method, via the perturbation operator theorem of Keller and
Liverani, has been exploited in recent papers to establish local limit and
Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov
chains. The main difficulty of this approach is to prove Taylor expansions for
the dominating eigenvalue of the Fourier kernels. This paper outlines this
method and extends it by proving a multi-dimensional local limit theorem, a
first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type
theorem in the sense of Prohorov metric. When applied to uniformly or
geometrically ergodic chains and to iterative Lipschitz models, the above cited
limit theorems hold under moment conditions similar, or close, to those of the
i.i.d. case.
http://arxiv.org/abs/0901.4617
Author(s): Jeffrey E. Steif
Abstract: Percolation is one of the simplest and nicest models in probability
theory/statistical mechanics which exhibits critical phenomena. Dynamical
percolation is a model where a simple time dynamics is added to the (ordinary)
percolation model. This dynamical model exhibits very interesting behavior. Our
goal in thissurvey is to give an overview of the work in dynamical percolation
that has been done (and some of which is in the process of being written up).
http://arxiv.org/abs/0901.4760
Author(s): Jeremie Unterberger (IECN)
Abstract: We construct in this article an explicit rough path over a multi-dimensional
fractional Brownian motion $B$ with arbitrary Hurst index $H$ (in particular,
for $H<1/4$) by regularizing an associated random Fourier series defined in
\cite{Unt08}. The regularization procedure is applied to 'Fourier normal
ordered' iterated integrals obtained by permuting the order of integration so
that innermost integrals have highest Fourier modes. The algebraic properties
of this rough path are best understood using the Hopf algebra structure of the
algebra of decorated rooted trees. Rough path theory gives then a general
procedure to define a stochastic calculus and solve stochastic differential
equations driven by this very irregular process. A variant of our
regularization scheme is also expected to apply to arbitrary deterministic
H\"older paths. The last section is also dedicated to the definition of a
related two-dimensional Gaussian process, called {\em antisymmetric
two-dimensional fractional Brownian motion}, with the same regularity as $B$
but with dependent components, to which the above construction extends
naturally.
http://arxiv.org/abs/0901.4771
Author(s): Rodrigo Bissacot and Eduardo Garibaldi
Abstract: Let $\sigma$: S -> S be the left shift acting on S, a one-sided Markov
subshift on a countable alphabet. Our intention is to guarantee the existence
of $\sigma$-invariant Borel probabilities that maximize the integral of a given
locally H\"older continuous potential A:S -> R. Under certain conditions, we
are able to show not only that A-maximizing probabilities do exist, but also
that they are characterized by the fact their support lies actually in a
particular Markov subshift on a finite alphabet. To that end, we make use of
objects dual to maximizing measures, the so-called sub-actions (concept
analogous to subsolutions of the Hamilton-Jacobi equation), and specially the
calibrated sub-actions (notion similar to weak KAM solutions).
http://arxiv.org/abs/0901.4640
Author(s): Yuri Yakubovich
Abstract: For a subfamily of multiplicative measures on integer partitions we give
conditions for properly rescaled associated Young diagrams to converge in
probability to a certain deterministic curve named the limit shape of
partitions. We provide explicit formulas for the scaling function and the limit
shape covering some known and some new examples.
http://arxiv.org/abs/0901.4655
Author(s): Joseph Najnudel and Ashkan Nikeghbali and Felix Rubin
Abstract: In this paper, we are interested in the asymptotic properties for the largest
eigenvalue of the Hermitian random matrix ensemble, called the Generalized
Cauchy ensemble $GCy$, whose eigenvalues PDF is given by
$$\textrm{const}\cdot\prod_{1\leq j-1/2$ and where $N$ is the size of the matrix ensemble. Using
results by Borodin and Olshanski \cite{Borodin-Olshanski}, we first prove that
for this ensemble, the largest eigenvalue divided by $N$ converges in law to
some probability distribution for all $s$ such that $\Re(s)>-1/2$. Using
results by Forrester and Witte \cite{Forrester-Witte2} on the distribution of
the largest eigenvalue for fixed $N$, we also express the limiting probability
distribution in terms of some non-linear second order differential equation.
Eventually, we show that the convergence of the probability distribution
function of the re-scaled largest eigenvalue to the limiting one is at least of
order $(1/N)$.
http://arxiv.org/abs/0901.4800
Author(s): Yaozhong Hu and Jia-an Yan
Abstract: This paper surveys some results on Wick product and Wick renormalization. The
framework is the abstract Wiener space. Some known results on Wick product and
Wick renormalization in the white noise analysis framework are presented for
classical random variables. Some conditions are described for random variables
whose Wick product or whose renormalization are integrable random variables.
Relevant results on multiple Wiener integrals, second quantization operator,
Malliavin calculus and their relations with the Wick product and Wick
renormalization are also briefly presented. A useful tool for Wick product is
the $S$-transform which is also described without the introduction of
generalized random variables.
http://arxiv.org/abs/0901.4911
Author(s): Yaozhong Hu and David Nualart
Abstract: We study a least squares estimator $\hat {\theta}_T$ for the
Ornstein-Uhlenbeck process, $dX_t=\theta X_t dt+\sigma dB^H_t$, driven by
fractional Brownian motion $B^H$ with Hurst parameter $H\ge \frac12$. We prove
the strong consistence of $\hat {\theta}_T$ (the almost surely convergence of
$\hat {\theta}_T$ to the true parameter ${% \theta}$). We also obtain the rate
of this convergence when $1/2\le H<3/4$, applying a central limit theorem for
multiple Wiener integrals. This least squares estimator can be used to study
other more simulation friendly estimators such as the estimator $\tilde
\theta_T$ defined by (4.1).
http://arxiv.org/abs/0901.4925
Author(s): Yevgeniy Kovchegov
Abstract: We derive an adiabatic theorem for Markov chains using well known facts about
mixing and relaxation times. We discuss the results in the context of the
recent developments in adiabatic quantum computation.
http://arxiv.org/abs/0901.4954
Author(s): Nizar Demni
Abstract: We express generalized Cauchy-Stieltjes transforms of some particular Beta
distributions (of ultraspherical type generating functions for orthogonal
polynomials) as a powered Cauchy-Stieltjes transform of some measure. For
suitable values of the power parameter, the latter measure turns out to be a
probability measure and its density is written down using Markov transforms.
The discarded values give a negative answer to a deformed free probability
unless a restriction on the power parameter is made. A particular symmetric
distribution interpolating between Wigner and arcsine distributions is
obtained. Its moments are expressed through a terminating hypergeometric series
interpolating between Catalan and shifed Catalan numbers. for small values of
the power parameter, the free cumulants are computed. Interesting opne problems
related to a deformed representation theory of the infinite symmetric group and
to a deformed Bozejko's convolution are discussed.
http://arxiv.org/abs/0902.0054
Author(s): Olga V. Aryasova and Andrey Yu. Pilipenko
Abstract: We consider a Brownian motion on the plane with semipermeable membranes on n
rays that have a common endpoint in the origin. We obtain the necessary and
sufficient conditions for the process to reach the origin and we show that the
probability of hitting the origin is equal to zero or one.
http://arxiv.org/abs/0902.0067
Author(s): Daniel Gentner and G\"unter Last
Abstract: We consider a lcsc group G acting properly on a Borel space S and measurably
on an underlying sigma-finite measure space. Our first main result is a
transport formula connecting the Palm pairs of jointly stationary random
measures on S. A key (and new) technical result is a measurable disintegration
of the Haar measure on G along the orbits. The second main result is an
intrinsic characterization of the Palm pairs of a G-invariant random measure.
We then proceed with deriving a general version of the mass-transport principle
for possibly non-transitive and non-unimodular group operations first in a
deterministic and then in its full probabilistic form.
http://arxiv.org/abs/0902.0068
Author(s): Itai Benjamini and Ori Gurel-Gurevich and Oded Schramm
Abstract: We construct a bounded degree graph G, such that a simple random walk on it
is transient but the random walk path (i.e., the subgraph of all the edges the
random walk has crossed) has only finitely many cutpoints, almost surely. We
also prove that the expected number of cutpoints of any transient Markov chain
is infinite. This answers two questions of James, Lyons and Peres.
Additionally, we consider a simple random walk on a finite connected graph G
that starts at some fixed vertex x and is stopped when it first visits some
other fixed vertex y. We provide a lower bound on the expected effective
resistance between x and y in the path of the walk, giving a partial answer to
a question raised in http://arxiv.org/abs/math/0603060
http://arxiv.org/abs/0902.0115
Author(s): Soumik Pal
Abstract: This principal result in this article is that every Poisson-Dirichlet
distribution PD(0,a) is an asymptotically invariant distribution for a growing
collection of independent Bessel square processes of dimension zero divided by
their total sum, under the condition that the sum total of their initial values
grows to infinity in probability. Implications in several areas of Probability
theory have been discussed, including Brownian local time, Fernholz &
Karatzas's Volatility Stabilized Market models of Mathematical Finance,
Watterson's Infinitely Many Neutral Alleles model in Statistical Genetics,
branching Bessel diffusions, and the Poisson-Dirichlet cascades. A key step
involves generalization of a polar decomposition result involving squared
Bessel processes that was observed by Warren & Yor in their study of the
Brownian burglar.
http://arxiv.org/abs/0902.0116
Author(s): Jean-Christophe Mourrat
Abstract: For the random walk among random conductances, we prove an algebraic decay of
the variance of a large class of functionals of the environment viewed by the
particle, our main hypothesis being that the conductances are bounded away from
zero. The basis of our method is the establishment of a Nash inequality,
followed either by a comparison with the simple random walk or by a more direct
analysis based on a martingale decomposition. As an example of application, we
show that under certain conditions, our results imply an estimate of the speed
of convergence of the mean square displacement of the walk towards its limit.
http://arxiv.org/abs/0902.0204
Author(s): Remco van der Hofstad
Abstract: We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. We show
that the critical behavior depends sensitively on the properties of the
asymptotic degrees. Indeed, when the proportion of vertices with degree at
least $k$ is bounded above by $k^{-\tau+1}$ for some $\tau>4$, the largest
critical connected component is of order $n^{2/3}$, where $n$ denotes the size
of the graph, as on the Erd\H{o}s-R\'enyi random graph. The restriction
$\tau>4$ corresponds to finite {\it third} moment of the degrees. When, the
proportion of vertices with degree at least $k$ is asymptotically equal to
$ck^{-\tau+1}$ for some $\tau\in (3,4),$ the largest critical connected
component is of order $n^{(\tau-2)/(\tau-1)},$ instead.
Our results show that, for inhomogeneous random graphs with a power-law
degree sequence, the critical behavior admits a transition when the third
moment of the degrees turns from finite to infinite. Similar phase transitions
have been shown to occur for typical distances in such random graphs when the
variance of the degrees turns from finite to infinite. We present further
results related to the size of the critical or scaling window, and state
conjectures for this and related random graph models.
http://arxiv.org/abs/0902.0216
Author(s): Krzysztof Szajowski and Mitsushi Tamaki
Abstract: A version of the secretary problem called the duration problem, in which the
objective is to maximize the time of possession of relatively best objects or
the second best, is treated. It is shown that in this duration problem there
are threshold numbers $(k_1^\star,k_2^\star)$ such that the optimal strategy
immediately selects a relatively best object if it appears after time
$k_1^\star$ and a relatively second best object if it appears after moment
$k_2^\star$. When number of objects tends to infinity the thresholds values are
$\lfloor 0.417188N\rfloor$ and $\rfloor 0.120381N\rfloor$, respectively. The
asymptotic mean time of shelf life of the object is $0.403827N$.
http://arxiv.org/abs/0902.0232
Author(s): Elizabeth S. Meckes
Abstract: The purpose of this paper is to synthesize the approaches taken by
Chatterjee-Meckes and Reinert-R\"ollin in adapting Stein's method of
exchangeable pairs for multivariate normal approximation. The more general
linear regression condition of Reinert-R\"ollin allows for wider applicability
of the method, while the method of bounding the solution of the Stein equation
due to Chatterjee-Meckes allows for improved convergence rates. Two abstract
normal approximation theorems are proved, one for use when the underlying
symmetries of the random variables are discrete, and one for use in contexts in
which continuous symmetry groups are present. The application to runs on the
line from Reinert-R\"ollin is reworked to demonstrate the improvement in
convergence rates, and a new application to joint value distributions of
eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian
manifold is presented.
http://arxiv.org/abs/0902.0333
Author(s): John Harnad and Alexander Yu. Orlov
Abstract: Tau functions expressed as fermionic expectation values are shown to provide
a natural and straightforward description of a number of random processes and
statistical models involving hard core configurations of identical particles on
the integer lattice, like a discrete version simple exclusion processes (ASEP),
nonintersecting random walkers, lattice Coulomb gas models and others, as well
as providing a powerful tool for combinatorial calculations involving paths
between pairs of partitions. We study the decay of the initial step function
within the discrete ASEP (d-ASEP) model as an example.
http://arxiv.org/abs/0704.1157
Author(s): Abdelmalek Abdesselam and Aldo Procacci and Benedetto Scoppola
Abstract: We prove clustering estimates for the truncated correlations, i.e., cumulants
of an unbounded spin system on the lattice. We provide a unified treatment,
based on cluster expansion techniques, of four different regimes: large mass,
small interaction between sites, large self-interaction, as well as the more
delicate small self-interaction or `low temperature' regime. A clustering
estimate in the latter regime is needed for the Bosonic case of the recent
result obtained by Lukkarinen and Spohn on the rigorous control on kinetic
scales of quantum fluids.
http://arxiv.org/abs/0901.4756
Author(s): Laszlo Lovasz
Abstract: In the last decade it became apparent that a large number of the most
interesting structures and phenomena of the world can be described by networks:
separable elements, with connections (or interactions) between certain pairs of
them.
These huge networks pose exciting challenges for the mathematician. Graph
Theory (the mathematical theory of networks) faces novel, unconventional
problems: these very large networks (like the Internet) are never completely
known, in most cases they are not even well defined. Data about them can be
collected only by indirect means like random local sampling.
Dense networks (in which a node is adjacent to a positive percent of others
nodes) and sparse networks (in which a node has a bounded number of neighbors)
show very different behavior. From a practical point of view, sparse networks
are more important, but at present we have more complete theoretical results
for dense networks. The paper surveys relations with probability, algebra,
extrema graph theory, and analysis.
http://arxiv.org/abs/0902.0132
Author(s): Persi Diaconis and Jason Fulman
Abstract: The "carries" when n random numbers are added base b form a Markov chain with
an "amazing" transition matrix determined by Holte. This same Markov chain
occurs in following the number of descents or rising sequences when n cards are
repeatedly riffle shuffled. We give generating and symmetric function proofs
and determine the rate of convergence of this Markov chain to stationarity.
Similar results are given for type B shuffles. We also develop connections with
Gaussian autoregressive processes and the Veronese mapping of commutative
algebra.
http://arxiv.org/abs/0902.0179
Author(s): Svante Janson
Abstract: We develop a theory of limits of finite posets in close analogy to the recent
theory of graph limits. In particular, we study representations of the limits
by functions of two variables on a probability space, and connections to
exchangeable random infinite posets.
http://arxiv.org/abs/0902.0306
Author(s): Aljosa Volcic
Abstract: In this paper we prove almost sure convergence to the ball, in the Nikodym
metric, of sequences of random Steiner symmetrizations of bounded Caccioppoli
and bounded measurable sets, paralleling a result due to Mani-Levitska
concerning convex bodies.
http://arxiv.org/abs/0902.0462
Author(s): Zbigniew Palmowski and Maria Vlasiou
Abstract: We consider a queuing model with the workload evolving between consecutive
i.i.d. exponential timers $\{e_q^{(i)}\}_{i=1,2,...}$ according to a spectrally
positive L\'{e}vy process $Y(t)$ which is reflected at 0. When the exponential
clock $e_q^{(i)}$ ends, the additional state-dependent service requirement
modifies the workload so that the latter is equal to $F_i(Y(e_q^{(i)}))$ at
epoch $e^{(1)}_q+...+e^{(i)}_q$ for some random nonnegative i.i.d. functionals
$F_i$. In particular, we focus on the case when $F_i(y)=(B_i-y)^+$, where
$\{B_i\}_{i=1,2,...}$ are i.i.d. nonnegative random variables. We analyse the
steady-state workload distribution for this model.
http://arxiv.org/abs/0902.0485
Author(s): Andreas Neuenkirch and Samy Tindel (IECN) and J\'er\'emie Unterberger (IECN)
Abstract: In this article, we give sharp bounds for the Euler- and trapezoidal
discretization of the Levy area associated to a d-dimensional fractional
Brownian motion. We show that there are three different regimes for the exact
root mean-square convergence rate of the Euler scheme. For H<3/4 the exact
convergence rate is n^{-2H+1/2}, where n denotes the number of the
discretization subintervals, while for H=3/4 it is n^{-1} (log(n))^{1/2} and
for H>3/4 the exact rate is n^{-1}. Moreover, the trapezoidal scheme has exact
convergence rate n^{-2H+1/2} for H>1/2. Finally, we also derive the asymptotic
error distribution of the Euler scheme. For H lesser than 3/4 one obtains a
Gaussian limit, while for H>3/4 the limit distribution is of Rosenblatt type.
http://arxiv.org/abs/0902.0497
Author(s): Carl Graham (CMAP)
Abstract: Multi-class systems having possibly both finite and infinite classes are
investigated under a natural partial exchangeability assumption. It is proved
that the conditional law of such a system, given the vector of the empirical
measures of its finite classes and directing measures of its infinite ones
(given by the de Finetti Theorem), corresponds to sampling independently from
each class, without replacement from the finite classes and i.i.d. from the
directing measure for the infinite ones. The equivalence between the
convergence of multi-exchangeable systems with fixed class sizes and the
convergence of the corresponding vectors of measures is then established.
http://arxiv.org/abs/0902.0539
Author(s): Florence Merlev\`ede (LAMA) and Magda Peligrad and Emmanuel Rio (LM-Versailles, INRIA Bordeaux - Sud-Ouest)
Abstract: In this paper we present a tail inequality for the maximum of partial sums of
a weakly dependent sequence of random variables that are not necessarily
bounded. The class considered includes geometrically and subgeometrically
strongly mixing sequences. The result is then used to derive asymptotic
moderate deviations results. Applications include classes of Markov chains,
functions of linear processes with absolutely regular innovations and ARCH
models
http://arxiv.org/abs/0902.0582
Author(s): J\'er\^ome Depauw (LMPT and FRDP) and Jean-Marc Derrien (LM-Brest)
Abstract: The Central Limit Theorem for the random walk on a stationary random network
of conductances has been studied by several authors. In one dimension, when
conductances and resistances are integrable, and following a method of
martingale introduced by S. Kozlov (1985), we can prove the Quenched Central
Limit Theorem. In that case the variance of the limit law is not null. When
resistances are not integrable, the Annealed Central Limit Theorem with null
variance was established by Y. Derriennic and M. Lin (personal communication).
The quenched version of this last theorem is proved here, by using a very
simple method. The similar problem for the continuous diffusion is then
considered. Finally our method allows us to prove an inequality for the
quadratic mean of a diffusion (X_t)_t at all time t.
http://arxiv.org/abs/0902.0584
Author(s): Justin Salez (INRIA Rocquencourt) and Devavrat Shah (MIT)
Abstract: The random assignment problem asks for the minimum-cost perfect matching in
the complete $n\times n$ bipartite graph $\Knn$ with i.i.d. edge weights, say
uniform on $[0,1]$. In a remarkable work by Aldous (2001), the optimal cost was
shown to converge to $\zeta(2)$ as $n\to\infty$, as conjectured by M\'ezard and
Parisi (1987) through the so-called cavity method. The latter also suggested a
non-rigorous decentralized strategy for finding the optimum, which turned out
to be an instance of the Belief Propagation (BP) heuristic discussed by Pearl
(1987). In this paper we use the objective method to analyze the performance of
BP as the size of the underlying graph becomes large. Specifically, we
establish that the dynamic of BP on $\Knn$ converges in distribution as
$n\to\infty$ to an appropriately defined dynamic on the Poisson Weighted
Infinite Tree, and we then prove correlation decay for this limiting dynamic.
As a consequence, we obtain that BP finds an asymptotically correct assignment
in $O(n^2)$ time only. This contrasts with both the worst-case upper bound for
convergence of BP derived by Bayati, Shah and Sharma (2005) and the best-known
computational cost of $\Theta(n^3)$ achieved by Edmonds and Karp's algorithm
(1972).
http://arxiv.org/abs/0902.0585
Author(s): Alain Camanes (LMJL)
Abstract: We consider a model of heat conduction networks consisting of oscillators in
contact with heat baths at different temperatures. Our aim is to generalize the
results concerning the existence and uniqueness of the stationnary state
already obtained when the network is reduced to a chain of particles. Using
Lasalle's principle, we establish a condition on the disposition of the heat
baths among the network that ensures the uniqueness of the invariant measure.
We will show that this condition is sharp when the oscillators are linear.
Moreover, when the interaction between the particles is stronger than the
pinning, we prove that this condition implies the existence of the invariant
measure.
http://arxiv.org/abs/0902.0586
Author(s): Louis-Pierre Arguin and Nicola Kistler
Abstract: We show through a simple example that perturbations of the Hamiltonian of a
spin glass which cannot be detected at the level of the free energy can
completely alter the behavior of the overlap. In particular, perturbations of
order O(log N), with N the size of the system, suffice to have ultrametricity
emerge in the thermodynamical limit.
http://arxiv.org/abs/0902.0294
Author(s): Ostap Hryniv and Yvan Velenik
Abstract: We introduce a class of models of semiflexible polymers. The latter are
characterized by a strong rigidity, the correlation length associated to the
gradient-gradient correlations, called the persistence length, being of the
same order as the polymer length.
We determine the macroscopic scaling limit, from which we deduce bounds on
the free energy of a polymer confined inside a narrow tube.
http://arxiv.org/abs/0902.0694
Author(s): Benjamin J. Fleming and Peter J. Forrester and Eric Nordenstam
Abstract: The bead process is the particle system defined on parallel lines, with
underlying measure giving constant weight to all configurations in which
particles on neighbouring lines interlace, and zero weight otherwise. Motivated
by the statistical mechanical model of the tiling of an $abc$-hexagon by three
species of rhombi, a finitized version of the bead process is defined. The
corresponding joint distribution can be realized as an eigenvalue probability
density function for a sequence of random matrices. The finitized bead process
is determinantal, and we give the correlation kernel in terms of Jacobi
polynomials. Two scaling limits are considered: a global limit in which the
spacing between lines goes to zero, and a certain bulk scaling limit. In the
global limit the shape of the support of the particles is determined, while in
the bulk scaling limit the bead process kernel of Boutillier is reclaimed,
after approriate identification of the anisotropy parameter therein.
http://arxiv.org/abs/0902.0709
Author(s): Remi Peyre
Abstract: We consider a stochastic N-particle model for the spatially homogeneous
Boltzmann evolution and show its convergence to the associated Boltzmann
equation when N tends to infinity. More precisely, for any time T>0 we bound
over the distance between the empirical measure of the particle system and the
measure given by Boltzmann evolution. That distance is computed in some
homogeneous Sobolev space. The control we get is Gaussian, i.e. we prove that
the distance is bigger than $x N^{-1/2}$ with a probability of type $e^{-x^2}$
at most. The two ingredients needed are first a control of fluctuations due to
the discrete nature of collisions, secondly a kind of Lipschitz continuity for
the Boltzmann collision kernel. The latter condition, in our present setting,
is only satisfied for Maxwellian models. We also have to control the initial
situation of the particle evolution, which we do by a kind of Chernoff
inequality for the i.i.d. case. Numerical applications tend to show that our
results are useful in practice.
http://arxiv.org/abs/0902.0721
Author(s): Nolwen Huet (IMT)
Abstract: We prove an isoperimetric inequality for probability measures $\mu$ on
$\mathbb{R}^n$ with density proportional to $\exp(-\phi(\lambda | x|))$, where
$|x|$ is the euclidean norm on $\mathbb{R}^n$ and $\phi$ is a non-decreasing
convex function. It applies in particular when $\phi(x)=x^\alpha$ with
$\alpha\ge1$. Under mild assumptions on $\phi$, the inequality is
dimension-free if $\lambda$ is chosen such that the covariance of $\mu$ is the
identity.
http://arxiv.org/abs/0902.0743
Author(s): Siva Athreya and Sreekar Vadlamani
Abstract: In this article we present an example of a random oriented tree model on
d-dimensional lattice, that is a forest in d=3 with positive probability. This
is in contrast with the other random tree models in the literature which are a
forest only when d strictly greater than 3.
http://arxiv.org/abs/0902.0762
Author(s): Craig A. Tracy and Harold Widom
Abstract: A limit theorem for the total current in the asymmetric simple exclusion
process (ASEP) with step initial condition is proved. This extends the result
of Johansson on TASEP to ASEP.
http://arxiv.org/abs/0902.0821
Author(s): A. D. Barbour
Abstract: The paper concerns the classical occupancy scheme with infinitely many boxes.
We establish approximations to the distributions of the number of occupied
boxes, and of the number of boxes containing exactly r balls, within the family
of translated Poisson distributions. These are shown to be of ideal asymptotic
order, with respect both to total variation distance and to the approximation
of point probabilities. The proof is probabilistic, making use of a translated
Poisson approximation theorem of R\"ollin (2005).
http://arxiv.org/abs/0902.0879
Author(s): Sanda N. Socoll and A. D. Barbour
Abstract: The paper is concerned with the equilibrium distributions of continuous-time
density dependent Markov processes on the integers. These distributions are
known typically to be approximately normal, and the approximation error, as
measured in Kolmogorov distance, is of the smallest order that is compatible
with their having integer support. Here, an approximation in the much stronger
total variation norm is established, without any loss in the asymptotic order
of accuracy; the approximating distribution is a translated Poisson
distribution having the same variance and (almost) the same mean. Our arguments
are based on the Stein-Chen method and Dynkin's formula.
http://arxiv.org/abs/0902.0884
Author(s): Sanda N. Socoll and A. D. Barbour
Abstract: The paper is concerned with the equilibrium distribution $\Pi_n$ of the
$n$-th element in a sequence of continuous-time density dependent Markov
processes on the integers. Under a $(2+\a)$-th moment condition on the jump
distributions, we establish a bound of order $O(n^{-(\a+1)/2}\sqrt{\log n})$ on
the difference between the point probabilities of $\Pi_n$ and those of a
translated Poisson distribution with the same variance. Except for the factor
$\sqrt{\log n}$, the result is as good as could be obtained in the simpler
setting of sums of independent integer-valued random variables. Our arguments
are based on the Stein-Chen method and coupling.
http://arxiv.org/abs/0902.0886
Author(s): Lorenz A. Gilch and Sebastian M\"uller
Abstract: Directed covers of finite graphs are also known as periodic trees or trees
with finitely many cone types. We expand the existing theory of directed covers
of finite graphs to those of infinite graphs. While the lower growth rate still
equals the branching number, upper and lower growth rate do not longer coincide
in general. Furthermore, the behaviour of random walks on directed covers of
infinite graphs is more subtile. We provide a classification in recurrence and
transience and point out that the critical random walk may be recurrent or
transient. Our proof is based on the observation that recurrence of the random
walk is equivalent to the almost sure extinction of an appropriate branching
process. Two examples in random environment are provided: homesick random walk
on infinite percolation clusters and random walk in random environment on
directed covers. Furthermore, we calculate, under reasonable assumptions, the
rate of escape with respect to suitable length functions and prove the
existence of the asymptotic entropy including an explicit formula which is also
a new result for directed covers of finite graphs. In particular, the
asymptotic entropy of random walks on directed covers of finite graphs is
positive if and only if the random walk is transient.
http://arxiv.org/abs/0902.0908
Author(s): M.L.Keptsyna and A.Le Breton and M.Viot
Abstract: Filtering problems with general exponential quadratic criteria are
investigated for Gauss-Markov processes. In this setting, the Linear
Exponential Gaussian and Risk-Sensitive filtering problems are solved and it is
shown that they may have different solutions.
http://arxiv.org/abs/0902.0940
Author(s): Deanna Needell
Abstract: The Kaczmarz method is an iterative algorithm for solving systems of linear
equations Ax=b. Theoretical convergence rates for this algorithm were largely
unknown until recently when work was done on a randomized version of the
algorithm. It was proved that for overdetermined systems, the randomized
Kaczmarz method converges with expected exponential rate, independent of the
number of equations in the system. Here we analyze the case where the system
Ax=b is corrupted by noise, so we consider the system where Ax is approximately
b + r where r is an arbitrary error vector. We prove that in this noisy
version, the randomized method reaches an error threshold dependent on the
matrix A with the same rate as in the error-free case. We provide examples
showing our results are sharp in the general context.
http://arxiv.org/abs/0902.0958
Author(s): Ross Pinsky
Abstract: Consider a simple random walk on the integers with the following transition
mechanism. At each site $x$, the probability of jumping to the right is
$\omega(x)\in[\frac12,1)$, until the first time the process jumps to the left
from site $x$, from which time onward the probability of jumping to the right
is $\frac12$. We investigate the transience/recurrence properties of this
process in both deterministic and stationary, ergodic environments
$\{\omega(x)\}_{x\in Z}$. In deterministic environments, we also study the
speed of the process.
http://arxiv.org/abs/0902.1026
Author(s): Arno B.J. Kuijlaars
Abstract: Multiple orthogonal polynomials are traditionally studied because of their
connections to number theory and approximation theory. In recent years they
were found to be connected to certain models in random matrix theory. In this
paper we introduce the notion of a multiple orthogonal polynomial ensemble (MOP
ensemble) and derive some of their basic properties. It is shown that Angelesco
and Nikishin systems give rise to MOP ensembles and that the equilibrium
problems that are associated with these systems have a natural interpretation
in the context of MOP ensembles.
http://arxiv.org/abs/0902.1058
Author(s): Michael Braverman
Abstract: We give conditions under which the tail probability of the supremum over unit
interval of a Levy process with light tail is equivalent to the tail of the
value of the process at the right endpoint.
http://arxiv.org/abs/0902.1075
Author(s): Alexander Drewitz and Alejandro F. Ram\'irez
Abstract: Recently Simenhaus proved that for any elliptic random walk in random
environment, transience in the neighborhood of a given direction is equivalent
to the a.s. existence of a deterministic asymptotic direction and to transience
in any direction in the open half space defined by this asymptotic direction.
Here we prove an improved version of this result and review some open problems.
http://arxiv.org/abs/0902.1115
Author(s): Qi Zhang and Huaizhong Zhao
Abstract: In this paper we develop a new weak convergence and compact embedding method
to study the existence and uniqueness of the
$L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{1}})\otimes
L_{\rho}^2({\mathbb{R}^{d}};{\mathbb{R}^{d}})$ valued solution of backward
stochastic differential equations with p-growth coefficients. Then we establish
the probabilistic representation of the weak solution of PDEs with p-growth
coefficients via corresponding BSDEs.
http://arxiv.org/abs/0902.1148
Author(s): Louigi Addario-Berry and Svante Janson and Colin McDiarmid
Abstract: The spread of a connected graph G was introduced by Alon Boppana and Spencer
(1998) and measures how tightly connected the graph is. It is defined as the
maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X
is uniformly distributed on $V(G)$. We investigate the spread of a variety of
random graphs, in particular the random regular graphs G(n,d), d >= 3, and
Erdos-Renyi random graphs G_{n,p} in the supercritical range p>1/n. We show
that if p=c/n with c>1 fixed then with high probability the spread is bounded,
and prove similar statements for G(n,d), d >= 3. We also prove lower bounds on
the spread in the barely supercritical case p-1/n = o(1). Finally, we show that
for d large the spread of G(n,d) becomes arbitrarily close to that of the
complete graph K_n.
http://arxiv.org/abs/0902.1156
Author(s): Qiang Zhen and Charles Knessl
Abstract: We consider the $M/G/1$ queue with a processor sharing server. We study the
conditional sojourn time distribution, conditioned on the customer's service
requirement, as well as the unconditional distribution, in various asymptotic
limits. These include large time and/or large service request, and heavy
traffic, where the arrival rate is only slightly less than the service rate.
Our results demonstrate the possible tail behaviors of the unconditional
distribution, which was previously known in the cases $G=M$ and $G=D$ (where it
is purely exponential). We assume that the service density decays at least
exponentially fast. We use various methods for the asymptotic expansion of
integrals, such as the Laplace and saddle point methods.
http://arxiv.org/abs/0902.1199
Author(s): Qiang Zhen and Charles Knessl
Abstract: We consider the $M/M/1$-PS queue with processor sharing. We study the
conditional sojourn time distribution of an arriving customer, conditioned on
the number of other customers present. A new formula is obtained for the
conditional sojourn time distribution, using a discrete Green's function. This
is shown to be equivalent to some classic results of Pollaczeck and Vaulot from
1946. Then various asymptotic limits are studied, including large time and/or
large number of customers present, and heavy traffic, where the arrival rate is
only slightly less than the service rate.
http://arxiv.org/abs/0902.1200
Author(s): O. Khorunzhiy and J.-F. Marckert
Abstract: Let D be a Dyck path chosen uniformly from the set of Dyck paths with 2n
steps. We prove that the sequence of the exponential moments of the maximum of
D normalized by the square root of n converges in the limit of infinite n, and
therefore is bounded uniformly in n. This result justifies corresponding
assumption used to prove certain estimates of high moments of large random
matrices.
http://arxiv.org/abs/0902.1229
Author(s): Laurent Carraro and Nicole El Karoui and Jan Obloj
Abstract: We study the class of Azema-Yor (AY) processes defined from a general
semimartingale with a continuous running supremum process. We show that they
arise as unique strong solutions of the Bachelier stochastic differential
equation which we prove is equivalent to the Drawdown equation. Solutions of
the latter have the drawdown property: they always stay above a given function
of their past supremum. We then show that any process which satisfies the
drawdown property is in fact an AY process. The proofs exploit group structure
of the set of AY processes, indexed by functions, which we introduce.
Further, we study in detail AY martingales defined from a non-negative local
martingale converging to zero at infinity. In particular, we construct AY
martingales with a given terminal law and this allows us to rediscover the AY
solution to the Skorokhod embedding problem.
Finally, we prove new optimal properties of AY martingales relative to
concave ordering of terminal laws of martingales.
http://arxiv.org/abs/0902.1328
Author(s): Marco Romito
Abstract: Under suitable assumptions of regularity and non-degeneracy on the covariance
of the driving additive noise, any Markov solution to the stochastic
Navier-Stokes equations has an associated generator of the diffusion and is the
unique solution to the corresponding martingale problem. Some elementary
examples are discussed to interpret these results.
http://arxiv.org/abs/0902.1402
Author(s): Marco Romito
Abstract: We prove existence of weak martingale solutions satisfying an almost sure
version of the energy inequality and which constitute a (almost sure) Markov
process.
http://arxiv.org/abs/0902.1407
Author(s): L\'aszl\'o Lov\'asz and Bal\'azs Szegedy
Abstract: In an earlier paper the authors proved that limits of convergent graph
sequences can be described by various structures, including certain 2-variable
real functions called graphons, random graph models satisfying certain
consistency conditions, and normalized, multiplicative and reflection positive
graph parameters. In this paper we show that each of these structures has a
related, relaxed version, which are also equivalent. Using this, we describe a
further structure equivalent to graph limits, namely probability measures on
countable graphs that are ergodic with respect to the group of permutations of
the nodes.
As an application, we prove an analogue of the Positivstellensatz for graphs:
We show that every linear inequality between subgraph densities that holds
asymptotically for all graphs has a formal proof in the following sense: it can
be approximated arbitrarily well by another valid inequality that is a "sum of
squares" in the algebra of partially labeled graphs.
http://arxiv.org/abs/0902.1327
Author(s): Kevin P. Costello
Abstract: If f(x_1, x_2, ..., x_n) is a polynomial dependent on a large number of
independent Bernoulli random variables, what can be said about the maximum
concentration of f on any single value? For linear polynomials, this reduces to
one version of the classical Littlewood-Offord problem: Given nonzero constants
a_1 through a_n, what is the maximum number of sums of the form +/- a_1 +/- a_2
+/-... +/- a_n which take on any single value? Here we consider the case where
f is either a bilinear form or a quadratic form. For the bilinear case, we show
that the only forms having concentration significantly larger than n^{-1} are
those which are in a certain sense very close to being degenerate. For the
quadratic case, we show that no form having many nonzero coefficients has
concentration significantly larger than n^{-1/2}. In both cases the results are
nearly tight.
http://arxiv.org/abs/0902.1538
Author(s): R\'emi Rhodes (CEREMADE)
Abstract: This paper deals with homogenization of second order divergence form
parabolic operators with locally stationary coefficients. Roughly speaking,
locally stationary coefficients have two evolution scales: both an almost
constant microscopic one and a smoothly varying macroscopic one. The
homogenization procedure aims to give a macroscopic approximation that takes
into account the microscopic heterogeneities. This paper follows "Diffusion in
a locally stationary random environment" (published in Probability Theory and
Related Fields) and improves this latter work by considering possibly
degenerate diffusion matrices. The geometry of the homogenized equation shows
that the particle is trapped in subspace of R^d.
http://arxiv.org/abs/0902.1586
Author(s): Robert O. Bauer
Abstract: We give a direct construction of the conformally invariant measure on
self-avoiding loops in Riemann surfaces (Werner measure) from chordal
$\text{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use
Schramm's formula to construct a measure on boundary bubbles encircling an
interior point. After establishing covariance properties for this bubble
measure, we apply these properties to obtain a measure on loops by integrating
measures on boundary bubbles. We calculate the distribution of the conformal
radius of boundary bubbles encircling an interior point and deduce from it
explicit upper and lower bounds for the loop measure.
http://arxiv.org/abs/0902.1626
Author(s): David Nualart and Lluis Quer-Sardanyons
Abstract: In this paper we establish lower and upper Gaussian bounds for the solutions
to the heat and wave equations driven by an additive Gaussian noise, using the
techniques of Malliavin calculus and recent density estimates obtained by
Nourdin and Viens. In particular, we deal with the one-dimensional stochastic
heat equation in $[0,1]$ driven by the space-time white noise, and the
stochastic heat and wave equations in $\mathbb{R}^d$ ($d\geq 1$ and $d\leq 3$,
respectively) driven by a Gaussian noise which is white in time and has a
general spatially homogeneous correlation.
http://arxiv.org/abs/0902.1849
Author(s): C\'edric Boutillier and B\'eatrice de Tili\`ere
Abstract: We study a large class of critical two-dimensional Ising models, namely
critical Z-invariant Ising models. Fisher [Fis66] introduced a correspondence
between the Ising model and the dimer model on a decorated graph, thus setting
dimer techniques as a powerful tool for understanding the Ising model. In this
paper, we give a full description of the dimer model corresponding to the
critical Z-invariant Ising model, consisting of explicit expressions which only
depend on the local geometry of the underlying isoradial graph. Our main result
is an explicit local formula for the inverse Kasteleyn matrix, in the spirit of
[Ken02], as a contour integral of the discrete exponential function of
[Mer01a,Ken02] multiplied by a local function. Using results of [BdT08] and
techniques of [dT07b,Ken02], this yields an explicit local formula for a
natural Gibbs measure, and a local formula for the free energy. As a corollary,
we recover Baxter's formula for the free energy of the critical Z-invariant
Ising model [Bax89], and thus a new proof of it. The latter is equal, up to a
constant, to the logarithm of the normalized determinant of the Laplacian
obtained in [Ken02].
http://arxiv.org/abs/0902.1882
Author(s): Peter Gacs and Mathieu Hoyrup (INRIA Lorraine - LORIA) and Cristobal Rojas (CREA)
Abstract: We extend the notion of randomness (in the version introduced by Schnorr) to
computable Probability Spaces and compare it to a dynamical notion of
randomness: typicality. Roughly, a point is typical for some dynamic, if it
follows the statistical behavior of the system (Birkhoff's pointwise ergodic
theorem). We prove that a point is Schnorr random if and only if it is typical
for every mixing computable dynamics. To prove the result we develop some tools
for the theory of computable probability spaces (for example, morphisms) that
are expected to have other applications.
http://arxiv.org/abs/0902.1939
Author(s): Robert Els\"asser and Thomas Sauerwald
Abstract: We introduce a new technique for bounding the cover time of random walks by
relating it to the runtime of randomized broadcast. In particular, we strongly
confirm for dense graphs the intuition of Chandra et al. \cite{CRRST97} that
"the cover time of the graph is an appropriate metric for the performance of
certain kinds of randomized broadcast algorithms". In more detail, our results
are as follows: For any graph $G=(V,E)$ of size $n$ and minimum degree
$\delta$, we have $\mathcal{R}(G)= \Oh(\frac{|E|}{\delta} \cdot \log n)$, where
$\mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time.
This bound is tight for binary trees and tight up to logarithmic factors for
many graphs including hypercubes, expanders and lollipop graphs. For any
$\delta$-regular (or almost $\delta$-regular) graph $G$ it holds that
$\mathcal{R}(G) = \Omega(\frac{\delta^2}{n} \cdot \frac{1}{\log n})$. Together
with our upper bound on $\mathcal{R}(G)$, this lower bound strongly confirms
the intuition of Chandra et al. for graphs with minimum degree $\Theta(n)$,
since then the cover time equals the broadcast time multiplied by $n$
(neglecting logarithmic factors). Conversely, for any $\delta$ we construct
almost $\delta$-regular graphs that satisfy $\mathcal{R}(G) = \Oh(\max
\{\sqrt{n},\delta \} \cdot \log^2 n)$. Since any regular expander satisfies
$\mathcal{R}(G) = \Theta(n)$, the strong relationship given above does not hold
if $\delta$ is polynomially smaller than $n$. Our bounds also demonstrate that
the relationship between cover time and broadcast time is much stronger than
the known relationships between any of them and the mixing time (or the closely
related spectral gap).
http://arxiv.org/abs/0902.1735
Author(s): Paul Bourgade
Abstract: We prove a multidimensional extension of Selberg's central limit theorem for
$\log\zeta$, in which non-trivial correlations appear. In particular, this
answers a question by Coram and Diaconis about the mesoscopic fluctuations of
the zeros of the Riemann zeta function.
Similar results are given in the context of random matrices from the unitary
group. This shows the correspondence $n \leftrightarrow \log t$ not only
between the dimension of the matrix and the height on the critical line, but
also, in a local scale, for small deviations from the critical axis or the unit
circle.
http://arxiv.org/abs/0902.1757
Author(s): Uriel Feige and Abraham D. Flaxman and Dan Vilenchik
Abstract: It is known that random k-CNF formulas have a so-called satisfiability
threshold at a density (namely, clause-variable ratio) of roughly 2^k\ln 2: at
densities slightly below this threshold almost all k-CNF formulas are
satisfiable whereas slightly above this threshold almost no k-CNF formula is
satisfiable. In the current work we consider satisfiable random formulas, and
inspect another parameter -- the diameter of the solution space (that is the
maximal Hamming distance between a pair of satisfying assignments).
It was previously shown that for all densities up to a density slightly below
the satisfiability threshold the diameter is almost surely at least roughly n/2
(and n at much lower densities). At densities very much higher than the
satisfiability threshold, the diameter is almost surely zero (a very dense
satisfiable formula is expected to have only one satisfying assignment). In
this paper we show that for all densities above a density that is slightly
above the satisfiability threshold (more precisely at ratio (1+ \eps)2^k \ln 2,
\eps=\eps(k) tending to 0 as k grows) the diameter is almost surely
O(k2^{-k}n). This shows that a relatively small change in the density around
the satisfiability threshold (a multiplicative (1 + \eps) factor), makes a
dramatic change in the diameter. This drop in the diameter cannot be attributed
to the fact that a larger fraction of the formulas is not satisfiable (and
hence have diameter 0), because the non-satisfiable formulas are excluded from
consideration by our conditioning that the formula is satisfiable.
http://arxiv.org/abs/0902.2012
Author(s): James B. Martin
Abstract: We consider a model of queues in discrete time, with batch services and
arrivals. The case where arrival and service batches both have Bernoulli
distributions corresponds to a discrete-time M/M/1 queue, and the case where
both have geometric distributions has also been previously studied. We describe
a common extension to a more general class where the batches are the product of
a Bernoulli and a geometric, and use reversibility arguments to prove versions
of Burke's theorem for these models. Extensions to models with continuous time
or continuous workload are also described. As an application, we show how these
results can be combined with methods of Seppalainen and O'Connell to provide
exact solutions for a new class of first-passage percolation problems.
http://arxiv.org/abs/0902.2026
Author(s): Arnaud Guillin and Christian Leonard (CMAP and MODAL'X) and Feng-Yu Wang and Liming Wu
Abstract: We continue our investigation on the transportation-information inequalities
$W_pI$ for a symmetric markov process, introduced and studied in \cite{GLWY}.
We prove that $W_pI$ implies the usual transportation inequalities $W_pH$, then
the corresponding concentration inequalities for the invariant measure $\mu$.
We give also a direct proof that the spectral gap in the space of Lipschitz
functions for a diffusion process implies $W_1I$ (a result due to \cite{GLWY})
and a Cheeger type's isoperimetric inequality. Finally we exhibit relations
between transportation-information inequalities and a family of functional
inequalities (such as $\Phi$-log Sobolev or $\Phi$-Sobolev).
http://arxiv.org/abs/0902.2101
Author(s): Tewfik Kernane
Abstract: We consider a single server retrial queue with the server subject to
interruptions and classical retrial policy for the access from the orbit to the
server. We analyze the equilibrium distribution of the system and obtain the
generating functions of the limiting distribution.
http://arxiv.org/abs/0902.2110
Author(s): Erika Hausenblas
Abstract: The aim of this note is to give some Burkholder-Davis-Gundy type inequalities
which are valid for the Ito stochastic integral with respect to Banach valued
Levy noise.
http://arxiv.org/abs/0902.2114
Author(s): Nicolas Bouleau (CERMA)
Abstract: It is possible to construct a double indexed process with sample paths a
surface of a family of subordinators obtained by subordination. We study here a
branch of this subordination process. This opens martingale methods on symbolic
calculus questions.
http://arxiv.org/abs/0902.2133
Author(s): Sebastian del Ba\~no Rollin and Albert Ferreiro-Castilla and Frederic Utzet
Abstract: A new expression for the characteristic function of log-spot in Heston model
is presented. This expression more clearly exhibits its properties as an
analytic characteristic function and allows us to compute the exact domain of
the moment generating function. This result is then applied to the volatility
smile at extreme strikes and to the control of the moments of spot. We also
give a factorization of the moment generating function as product of Bessel
type factors, and an approximating sequence to the law of log-spot is deduced.
http://arxiv.org/abs/0902.2154
Author(s): Seva Shneer and Vitali Wachtel
Abstract: For families of random walks $\{S_k^{(a)}\}$ with $\mathbf E S_k^{(a)} = -ka
< 0$ we consider their maxima $M^{(a)} = \sup_{k \ge 0} S_k^{(a)}$. We
investigate the asymptotic behaviour of $M^{(a)}$ as $a \to 0$ for
asymptotically stable random walks. This problem appeared first in the 1960's
in the analysis of a single-server queue when the traffic load tends to 1 and
since then is referred to as the heavy-traffic approximation problem. Kingman
and Prokhorov suggested two different approaches which were later followed by
many authors. We give two elementary proofs of our main result, using each of
these approaches. It turns out that the main technical difficulties in both
proofs are rather similar and may be resolved via a generalisation of the
Kolmogorov inequality to the case of an infinite variance. Such a
generalisation is also obtained in this note.
http://arxiv.org/abs/0902.2185
Author(s): Zhao Guo-xi and Hu Qi-Zhou
Abstract: The performance of non-preemptive M/M/1 queueing system with two priority is
analyzed. By using complementary variable method to make vector Markov process
and analyzing the state-change equations of the queueing system, the generating
function of two kinds of customers'length distribution are derived under
non-preemptive priority .Through further discussion, the probability of the
server that it is working or free and average length of two kinds of customers
are also derived.
http://arxiv.org/abs/0902.2086
Author(s): Rishi Talreja and Josh Reed
Abstract: We study the $G/GI/\infty$ queue from two different perspectives in the same
heavy-traffic regime. First, we represent the dynamics of the system using a
measure-valued process that keeps track of the age of each customer in the
system. Using the continuous-mapping approach together with the martingale
functional central limit theorem, we obtain fluid and diffusion limits for this
process in a space of distribution-valued processes. Next, we study a
measure-valued process that keeps track of the residual service time of each
customer in the system. In this case, using the functional central limit
theorem and the random time change theorem together with the continuous-mapping
approach, we again obtain fluid and diffusion limits in our space of
distribution-valued processes. In both cases, we find that our diffusion limits
may be characterized as distribution-valued Ornstein-Uhlenbeck processes.
Further, these diffusion limits can be analyzed using standard results from the
theory of Markov processes.
http://arxiv.org/abs/0902.2236
Author(s): Ecaterina Sava
Abstract: The main goal of this paper is to determine the Poisson boundary of
lamplighter random walks over a general class of discrete groups $\Gamma$
endowed with a rich boundary. The starting point is the Strip Criterion of
identification of the Poisson boundary for random walks on discrete groups due
to Kaimanovich. A geometrical method for constructing the strip as a subset of
the lamplighter group starting with a smaller strip in the base group $\Gamma$
is developed. Then, this method is applied to several classes of base groups
$\Gamma$: groups with infinitely many ends, hyperbolic groups in the sense of
Gromov, and Euclidean lattices. We show that under suitable hypothesis the
Poisson boundary for a class of random walks on lamplighter groups is the space
of infinite limit configurations.
http://arxiv.org/abs/0902.2285
Author(s): S. N. Ethier and Jiyeon Lee
Abstract: That two losing games can be combined to form a winning game is known as
Parrondo's paradox. We establish a strong law of large numbers and a central
limit theorem for the Parrondo player's sequence of profits, both in a
one-parameter family of profit-dependent games and in a two-parameter family of
history-dependent games, with the potentially winning game being either a
random mixture or a nonrandom pattern of the two losing games. We derive
formulas for the mean and variance parameters of the central limit theorem in
nearly all such scenarios; formulas for the mean permit an analysis of when the
Parrondo effect is present.
http://arxiv.org/abs/0902.2368
Author(s): Eran Shmaya
Abstract: An infinite two-player zero-sum game with a Borel winning set, in which the
opponent's actions are monitored eventually but not necessarily immediately
after they are played, admits a value. The proof relies on a representation of
the game as a stochastic game with perfect information, in which Nature
operates as a delegate for the players and performs the randomizations for
them.
http://arxiv.org/abs/0902.2254
Author(s): Mark W. Meckes
Abstract: This paper considers random (non-Hermitian) circulant matrices, and proves
several results analogous to recent theorems on non-Hermitian random matrices
with independent entries. In particular, the limiting spectral distribution of
a random circulant matrix is shown to be complex normal, and bounds are given
for the probability that a circulant sign matrix is singular.
http://arxiv.org/abs/0902.2472
Author(s): Tai Melcher
Abstract: This paper studies Brownian motion and heat kernel measure on a class of
infinite dimensional Lie groups. We prove a Cameron-Martin type
quasi-invariance theorem for the heat kernel measure and give estimates on the
$L^p$ norms of the Radon-Nikodym derivatives. We also prove that a logarithmic
Sobolev inequality holds in this setting.
http://arxiv.org/abs/0902.2500
Author(s): Harald Luschgy and Gilles Pag\`es (PMA)
Abstract: We derive a precise link between series expansions of Gaussian random vectors
in a Banach space and Parseval frames in their reproducing kernel Hilbert
space. The results are applied to pathwise continuous Gaussian processes and a
new optimal expansion for fractional Ornstein-Uhlenbeck processes is derived.
In the end an extension of this result to Gaussian stationary processes with
convex covariance function is established.
http://arxiv.org/abs/0902.2563
Author(s): Sebastian Jaimungal and Alex Krenin and and Angelo Valov
Abstract: The first passage time problem for Brownian motions hitting a barrier has
been extensively studied in the literature. In particular, many incarnations of
integral equations which link the density of the hitting time to the equation
for the barrier itself have appeared. Most interestingly, Peskir(2002b)
demonstrates that a master integral equation can be used to generate a
countable number of new equations via differentiation or integration by parts.
In this article, we generalize Peskir's results and provide a more powerful
unifying framework for generating integral equations through a new class of
martingales. We obtain a continuum of Volterra type integral equations of the
first kind and prove uniqueness for a subclass. Furthermore, through the
integral equations, we demonstrate how certain functional transforms of the
boundary affect the density function. Finally, we demonstrate a fundamental
connection between the Volterra integral equations and a class of Fredholm
integral equations.
http://arxiv.org/abs/0902.2569
Author(s): O.L.V. Costa and F. Dufour
Abstract: The main goal of this paper is to apply the so-called policy iteration
algorithm (PIA) for the long run average continuous control problem of
piecewise deterministic Markov processes (PDMP's) taking values in a general
Borel space and with compact action space depending on the state variable. In
order to do that we first derive some important properties for a pseudo-Poisson
equation associated to the problem. In the sequence it is shown that the
convergence of the PIA to a solution satisfying the optimality equation holds
under some classical hypotheses and that this optimal solution yields to an
optimal control strategy for the average control problem for the
continuous-time PDMP in a feedback form.
http://arxiv.org/abs/0902.2673
Author(s): Marc Arnaudon (LMA) and Anton Thalmaier
Abstract: In this paper we use methods from Stochastic Analysis to establish Li-Yau
type estimates for positive solutions of the heat equation. In particular, we
want to emphasize that Stochastic Analysis provides natural tools to derive
local estimates in the sense that the gradient bound at given point depends
only on universal constants and the geometry of the Riemannian manifold locally
about this point.
http://arxiv.org/abs/0902.2681
Author(s): Rainer Buckdahn (LM) and Boubakeur Labed and Catherine Rainer (LM) and Lazhar Tamer
Abstract: We consider a stochastic control problem which is composed of a controlled
stochastic differential equation, and whose associated cost functional is
defined through a controlled backward stochastic differential equation. Under
appropriate convexity assumptions on the coefficients of the forward and the
backward equations we prove the existence of an optimal control on a suitable
reference stochastic system. The proof is based on an approximation of the
stochastic control problem by a sequence of control problems with smooth
coefficients, admitting an optimal feedback control. The quadruplet formed by
this optimal feedback control and the associated solution of the forward and
the backward equations is shown to converge in law, at least along a
subsequence. The convexity assumptions on the coefficients then allow to
construct from this limit an admissible control process which, on an
appropriate reference stochastic system, is optimal for our stochastic control
problem.
http://arxiv.org/abs/0902.2693
Author(s): Erhan Bayraktar and Hao Xing
Abstract: The value function of an optimal stopping problem for a process with Levy
jumps is known to be a generalized solution of a variational inequality.
Assuming the diffusion component of the process is non-degenerate and a mild
assumption on the singularity of the Levy measure, this paper shows that the
value function is smooth in the continuation region for problems with either
finite or infinite variation jumps. Moreover, the smooth-fit property is shown
via the global regularity of the value function. This paper confirms the
intuition that the non-degenerate diffusion component dictates the regularity
of the value function in the optimal stopping problem for jump processes.
http://arxiv.org/abs/0902.2479
Author(s): Mike Ludkovski
Abstract: We study the numerical solution of nonlinear partially observed optimal
stopping problems. The system state is taken to be a multi-dimensional
diffusion and drives the drift of the observation process, which is another
multi-dimensional diffusion with correlated noise. Such models where the
controller is not fully aware of her environment are of interest in applied
probability and financial mathematics. We propose a new approximate numerical
algorithm based on the particle filtering and regression Monte Carlo methods.
The algorithm maintains a continuous state-space and yields an integrated
approach to the filtering and control sub-problems. Our approach is entirely
simulation-based and therefore allows for a robust implementation with respect
to model specification. We carry out the error analysis of our scheme and
illustrate with several computational examples. An extension to discretely
observed stochastic volatility models is also considered.
http://arxiv.org/abs/0902.2518
Author(s): Tobias Fritz
Abstract: This note gives generators and relations for the strict monoidal category of
probabilistic maps on finite cardinals (i.e., stochastic matrices).
http://arxiv.org/abs/0902.2554
Author(s): Kilian Raschel
Abstract: Nearest neighbor random walks in the quarter plane that are absorbed when
reaching the boundary are studied. The cases of positive and zero drift are
considered. Absorption probabilities at a given time and at a given site are
made explicit. The following asymptotics for these random walks starting from a
given point $(n_0,m_0)$ are computed : that of probabilities of being absorbed
at a given site $(i,0)$ [resp. $(0,j)$] as $i\to \infty$ [resp. $j \to
\infty$], that of the distribution's tail of absorption time at x-axis [resp.
y-axis], that of the Green functions at site $(i,j)$ when $i,j\to \infty$ and
$j/i \to \tan \gamma$ for $\gamma \in [0, \pi/2]$. These results give the
Martin boundary of the process and in particular the suitable Doob
$h$-transform in order to condition the process never to reach the boundary.
They also show that this $h$-transformed process is equal in distribution to
the limit as $n\to \infty$ of the process conditioned by not being absorbed at
time $n$. The main tool used here is complex analysis.
http://arxiv.org/abs/0902.2785
Author(s): M. Cranston and L. Koralov and S. Molchanov and B. Vainberg
Abstract: We consider the model for the distribution of a long homopolymer in a
potential field. The typical shape of the polymer depends on the temperature
parameter. We show that at a critical value of the temperature the transition
occurs from a globular to an extended phase. For various values of the
temperature, including those at or near the critical value, we consider the
limiting behavior of the polymer when its size tends to infinity.
http://arxiv.org/abs/0902.2830
Author(s): Julien Barral and Benoit Mandelbrot
Abstract: Statistically self-similar measures on $[0,1]$ are limit of multiplicative
cascades of random weights distributed on the $b$-adic subintervals of $[0,1]$.
These weights are i.i.d, positive, and of expectation $1/b$. We extend these
cascades naturally by allowing the random weights to take negative values. This
yields martingales taking values in the space of continuous functions on
$[0,1]$. Specifically, we consider for each $H\in (0,1)$ the martingale
$(B_{n})_{n\geq1}$ obtained when the weights take the values $-b^{-H}$ and
$b^{-H}$, in order to get $B_n$ converging almost surely uniformly to a
statistically self-similar function $B$ whose H\"{o}lder regularity and fractal
properties are comparable with that of the fractional Brownian motion of
exponent $H$. This indeed holds when $H\in(1/2,1)$. Also the construction
introduces a new kind of law, one that it is stable under random weighted
averaging and satisfies the same functional equation as the standard symmetric
stable law of index $1/H$. When $H\in(0,1/2]$, to the contrary, $B_n$ diverges
almost surely. However, a natural normalization factor $ a_n$ makes the
normalized correlated random walk $ B_n / a_n$ converge in law, as $n$ tends to
$\infty$, to the restriction to $[0,1]$ of the standard Brownian motion. Limit
theorems are also associated with the case $H>1/2$.
http://arxiv.org/abs/0902.2902
Author(s): Ion Nechita (ICJ) and Cl\'ement Pellegrini
Abstract: We consider a generalized model of repeated quantum interactions, where a
system $\mathcal{H}$ is interacting in a random way with a sequence of
independent quantum systems $\mathcal{K}_n, n \geq 1$. Two types of randomness
are studied in detail. One is provided by considering Haar-distributed
unitaries to describe each interaction between $\mathcal{H}$ and
$\mathcal{K}_n$. The other involves random quantum states describing each copy
$\mathcal{K}_n$. In the limit of a large number of interactions, we present
convergence results for the asymptotic state of $\mathcal{H}$. This is achieved
by studying spectral properties of (random) quantum channels which guarantee
the existence of unique invariant states. Finally this allows to introduce a
new physically motivated ensemble of random density matrices called the
\emph{asymptotic induced ensemble}.
http://arxiv.org/abs/0902.2634
Author(s): Yaming Yu
Abstract: Let K_n denote the smaller mode of the nth row of Stirling numbers of the
second kind S(n, k). Using a probablistic argument, it is shown that for all
n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of
x, and w(n) is Lambert's W-function.
http://arxiv.org/abs/0902.2964
Author(s): Ping He and Jiangang Ying
Abstract: In this paper, we shall prove that the irreducibility in the sense of fine
topology implies the uniqueness of invariant probability measures. It is also
proven that this irreducibility is strictly weaker than the strong Feller
property plus irreducibility in the sense of original topology, which is the
usual uniqueness condition.
http://arxiv.org/abs/0902.3296
Author(s): Freddy Delbaen (Department of Mathematics) and Ying Hu (IRMAR) and Xiaobo Bao (Department of Mathematics)
Abstract: In this paper, we discuss the solvability of backward stochastic differential
equations (BSDEs) with superquadratic generators. We first prove that given a
superquadratic generator, there exists a bounded terminal value, such that the
associated BSDE does not admit any bounded solution. On the other hand, we
prove that if the superquadratic BSDE admits a bounded solution, then there
exist infinitely many bounded solutions for this BSDE. Finally, we prove the
existence of a solution for Markovian BSDEs where the terminal value is a
bounded continuous function of a forward stochastic differential equation.
http://arxiv.org/abs/0902.3316
Author(s): M. Jara and C. Landim and A. Teixeira
Abstract: Fix a strictly positive measure $W$ on the $d$-dimensional torus
$\bb T^d$. For an integer $N\ge 1$, denote by $W^N_x$, $x=(x_1,
..., x_d)$, $0\le x_i 1$, if $W$ is a finite discrete measure, $W=\sum_{i\ge 1} w_i
\delta_{x_i}$, we prove that the random walk which jumps from $x/N$ uniformly
to one of its neighbors at rate $(W^N_x)^{-1}$ has a metastable behavior, as
defined in \cite{bl1}, described by the $K$-process introduced in \cite{fm1}.
http://arxiv.org/abs/0902.3334
Author(s): Grigori Olshanski
Abstract: We construct a family of Markov processes with continuous sample trajectories
on an infinite-dimensional space, the Thoma simplex. The family depends on
three continuous parameters, one of which, the Jack parameter, is similar to
the beta parameter in random matrix theory. The processes arise in a scaling
limit transition from certain finite Markov chains, the so called up-down
chains on the Young graph with the Jack edge multiplicities. Each of the limit
Markov processes is ergodic and its stationary distribution is a symmetrizing
measure. The infinitesimal generators of the processes are explicitly computed;
viewed as selfadjoint operators in the L^2 spaces over the symmetrizing
measures, the generators have purely discrete spectrum which is explicitly
described.
For the special value 1 of the Jack parameter, the limit Markov processes
coincide with those of the recent work by Borodin and the author (Prob. Theory
Rel. Fields 144 (2009), 281--318; arXiv:0810.3751). In the limit as the Jack
parameter goes to 0, our family of processes degenerates to the one-parameter
family of diffusions on the Kingman simplex studied long ago by Ethier and
Kurtz in connection with some models of population genetics.
The techniques of the paper are essentially algebraic. The main computations
are performed in the algebra of shifted symmetric functions with the Jack
parameter and rely on the concept of anisotropic Young diagrams due to Kerov.
http://arxiv.org/abs/0902.3395
Author(s): Carl Mueller and Leonid Mytnik and Jeremy Quastel
Abstract: We consider reaction-diffusion equations of KPP type in one spatial
dimension, perturbed by a Fisher-Wright white noise, under the assumption of
uniqueness in distribution. Examples include the randomly perturbed Fisher-KPP
equations $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u(1-u)}\dot
W, $ and $ \partial_t u = \partial_x^2 u + u(1-u) + \epsilon \sqrt{u}\dot W, $
where $\dot W= \dot W(t,x)$ is a space-time white noise.
We prove the Brunet-Derrida conjecture that the speed of traveling fronts is
asymptotically $ 2-\pi^2 |\log \epsilon^2|^{-2} $ up to a factor of order $
(\log|\log\epsilon|)|\log\epsilon|^{-3}$.
http://arxiv.org/abs/0902.3423
Author(s): Alexander I. Bufetov
Abstract: An asymptotic expansion is established for time averages of translation flows
on flat surfaces. This result, which extends earlier work of A.Zorich and
G.Forni, yields limit theorems for translation flows.
The argument, close in spirit to that of G.Forni, uses the approximation of
ergodic integrals by holonomy-invariant Hoelder cocycles on trajectories of the
flows. The space of holonomy-invariant Hoelder cocycles is finite-dimensional,
and is given by an explicit construction. First, a symbolic representation for
a uniquely ergodic translation flow is obtained following S.Ito and A.M.
Vershik, and then, the space of cocycles is constructed using a family of
finitely-additive complex-valued holonomy-invariant measures on the asymptotic
foliations of a Markov compactum.
http://arxiv.org/abs/0902.3303
Author(s): Michael Caruana and Peter Friz and Harald Oberhauser
Abstract: In a series of papers, starting with [Fully nonlinear stochastic partial
differential equations. C. R. Acad. Sci. Paris Ser. I Math. 326 (1998), no. 9]
Lions and Souganidis proposed a (pathwise) theory for fully non-linear
stochastic partial differential equations. We present here a (partial)
extension towards certain spatial dependence in the noise term. The main
novelty is the use of rough path theory in this context [Lyons, Terry J.;
Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14
(1998), no. 2, 215-310].
http://arxiv.org/abs/0902.3352
Author(s): Martin Hairer and Charles Manson
Abstract: We consider a one-dimensional diffusion process with coefficients that are
periodic outside of a finite 'interface region'. The question investigated in
this article is the limiting long time / large scale behaviour of such a
process under diffusive rescaling. Our main result is that it converges weakly
to a rescaled version of skew Brownian motion, with parameters that can be
given explicitly in terms of the coefficients of the original diffusion.
Our method of proof relies on the framework provided by Freidlin and Wentzell
for diffusion processes on a graph in order to identify the generator of the
limiting process. The graph in question consists of one vertex representing the
interface region and two infinite segments corresponding to the regions on
either side.
http://arxiv.org/abs/0902.3471
Author(s): Hirofumi Osada
Abstract: We investigate the construction of diffusions consisting of infinitely
numerous Brownian particles moving in $ \Rd $ and interacting via logarithmic
functions (2D Coulomb potentials). These potentials are really strong and long
range in nature. The associated equilibrium states are no longer Gibbs
measures.
We present general results for the construction of such diffusions and, as
applications thereof, construct two typical interacting Brownian motions with
logarithmic interaction potentials, namely the Dyson model in infinite
dimensions and Ginibre interacting Brownian motions. The former is a particle
system in $ \R $ while the latter is in $ \R ^2 $. Both models are translation
and rotation invariant in space, and as such, are prototypes of dimensions $ d
= 1,2 $, respectively. The equilibrium states of the former diffusion model are
determinantal random point fields with sine kernels. They appear in the
thermodynamical limits of the spectrum of the ensembles of Gaussian random
matrices such as GOE, GUE and GSE. The equilibrium states of the latter
diffusion model are the thermodynamical limits of the spectrum of the ensemble
of complex non-Hermitian Gaussian random matrices known as the Ginibre
ensemble.
http://arxiv.org/abs/0902.3561
Author(s): Kevin Fleming and Nicholas Pippenger
Abstract: We study random surfaces constructed by glueing together $N/k$ filled
$k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)...3\cdot 1$ pairings
of the edges being equally likely. (We assume that lcm $\{2,k\}$ divides $N$.)
The Euler characteristic of the resulting surface is related to the number of
cycles in a certain random permutation of $\{1, ..., N\}$. Gamburd has shown
that when 2 lcm $\{2,k\}$ divides $N$, the distribution of this random
permutation converges to that of the uniform distribution on the alternating
group $A_N$ in the total-variation distance as $N\to\infty$. We obtain
large-deviations bounds for the number of cycles that, together with Gamburd's
result, allow us to derive sharp estimates for the moments of the number of
cycles. These estimates allow us to confirm certain cases of conjectures made
by Pippenger and Schleich.
http://arxiv.org/abs/0902.3646
Author(s): Ellen Baake and Inke Herms
Abstract: Populations evolving under the joint influence of recombination and
resampling (traditionally known as genetic drift) are investigated. First, we
summarise and adapt a deterministic approach, as valid for infinite
populations, which assumes continuous time and single crossover events. The
corresponding nonlinear system of differential equations permits a closed
solution, both in terms of the type frequencies and via linkage disequilibria
of all orders. To include stochastic effects, we then consider the
corresponding finite-population model, the Moran model with single crossovers,
and examine it both analytically and by means of simulations. Particular
emphasis is on the connection with the deterministic solution. If there is only
recombination and every pair of recombined offspring replaces their pair of
parents (i.e., there is no resampling), then the {\em expected} type
frequencies in the finite population, of arbitrary size, equal the type
frequencies in the infinite population. If resampling is included, the
stochastic process converges, in the infinite-population limit, to the
deterministic dynamics, which turns out to be a good approximation already for
populations of moderate size.
http://arxiv.org/abs/q-bio/0612024
Author(s): Amin Coja-Oghlan
Abstract: Let F be a uniformly distributed random k-SAT formula with n variables and m
clauses. We present a polynomial time algorithm that finds a satisfying
assignment of F with high probability for constraint densities
m/n<(1-eps_k)2^k\ln(k)/k, where eps_k->0. Previously no efficient algorithm was
known to find solutions with non-vanishing probability beyond m/n=1.817.2^k/k
[Frieze and Suen, J. of Algorithms 1996].
http://arxiv.org/abs/0902.3583
Author(s): Thomas Duquesne and Jean-Francois Le Gall
Abstract: We prove a strong form of the invariance under re-rooting of the distribution
of the continuous random trees called Levy trees. This extends previous results
due to several authors.
http://arxiv.org/abs/0902.3735
Author(s): Xiaoyu Hu and Jason Miller and Yuval Peres
Abstract: Let $U \subseteq \C$ be a bounded domain with smooth boundary and let $F$ be
an instance of the continuum Gaussian free field on $U$ with respect to the
Dirichlet inner product $\int_U \nabla f(x) \cdot \nabla g(x) dx$. The set
$T(a;U)$ of $a$-thick points of $F$ consists of those $z \in U$ such that the
average of $F$ on a disk of radius $r$ centered at $z$ has growth $\sqrt{a/\pi}
\log \tfrac{1}{r}$ as $r \to 0$. We show that for each $0 \leq a \leq 2$ the
Hausdorff dimension of $T(a;U)$ is almost surely $2-a$ and that with
probability one $T(a;U)$ is empty when $a > 2$. Furthermore, we prove that
$T(a;U)$ is invariant under conformal transformations in an appropriate sense.
The notion of a thick point is connected to the Liouville quantum gravity
measure with parameter $\gamma$ given formally by $\Gamma(dz) = e^{\sqrt{2\pi}
\gamma F(z)} dz$ considered by Duplantier and Sheffield.
http://arxiv.org/abs/0902.3842
Author(s): Folkmar Bornemann
Abstract: We give a short, operator-theoretic proof of the asymptotic independence of
the minimal and maximal eigenvalue of the n \times n Gaussian Unitary Ensemble
in the large matrix limit n \to \infty. This is done by representing the joint
probability distribution of those extreme eigenvalues as the Fredholm
determinant of an operator matrix that asymptotically becomes diagonal. The
method is amenable to explicitly establish the leading order term of an
asymptotic expansion. As a corollary we obtain that the correlation of the
extreme eigenvalues asymptotically behaves like n^{-2/3}/4\sigma^2, where
\sigma^2 denotes the variance of the Tracy--Widom distribution.
http://arxiv.org/abs/0902.3870
Author(s): Patricia Goncalves
Abstract: We prove a Central Limit Theorem for the empirical measure in the
one-dimensional Totally Asymmetric Zero-Range Process in the hyperbolic scaling
$N$, starting from the equilibrium measure $\nu_{\rho}$. We also show that when
taking the direction of the characteristics, the limit density fluctuation
field does not evolve in time until $N^{4/3}$, which implies the current across
the characteristics to vanish in this longer time scale.
http://arxiv.org/abs/0902.3974
Author(s): Florian Lipsmeier and Ellen Baake
Abstract: The problem of \emph{statistical recognition} is considered, as it arises in
immunobiology, namely, the discrimination of foreign antigens against a
background of the body's own molecules. The precise mechanism of this
foreign-self-distinction, though one of the major tasks of the immune system,
continues to be a fundamental puzzle. Recent progress has been made by van den
Berg, Rand, and Burroughs (2001), who modelled the \emph{probabilistic} nature
of the interaction between the relevant cell types, namely, T-cells and
antigen-presenting cells (APCs). Here, the stochasticity is due to the random
sample of antigens present on the surface of every APC, and to the random
receptor type that characterises individual T-cells. It has been shown
previously that this model, though highly idealised, is capable of reproducing
important aspects of the recognition phenomenon, and of explaining them on the
basis of stochastic rare events. These results were obtained with the help of a
refined large deviation theorem and were thus asymptotic in nature. Simulations
have, so far, been restricted to the straightforward simple sampling approach,
which does not allow for sample sizes large enough to address more detailed
questions. Building on the available large deviation results, we develop an
importance sampling technique that allows for a convenient exploration of the
relevant tail events by means of simulation. With its help, we investigate the
mechanism of statistical recognition in some depth. In particular, we
illustrate how a foreign antigen can stand out against the self background if
it is present in sufficiently many copies, although no \emph{a priori}
difference between self and nonself is built into the model.
http://arxiv.org/abs/0901.2227
Author(s): Piotr Garbaczewski
Abstract: We analyze Levy flights subject to an influence of external potentials and/or
external conservative forces. Our goal is to clarify a discord between two
classes of pertinent processes: those driven by Langevin equation with Levy
noise and those named topological processes. Jump intensities of the latter
processes are locally modified (via multiplicative Gibbs-type factors) by a
"potential landscape" traveled by the flight and no explicit external forces
are used to modify (confine) the noise. The discussion is set within the
general framework of so-called Schrodinger boundary data problem which
encompasses both Gaussian and non-Gaussian Markov processes.
http://arxiv.org/abs/0902.3536
Author(s): Viktor P. Zastavnyi and Emilio Porcu
Abstract: We characterize completely the Gneiting class of space-time covariance
functions and give more relaxed conditions on the involved functions. We then
show necessary conditions for the construction of compactly supported functions
of the Gneiting type. These conditions are very general since they do not
depend on the Euclidean norm. Finally, we discuss a general class of positive
definite functions, used for multivariate Gaussian random fields. For this
class, we show necessary criteria for its generator to be compactly supported.
http://arxiv.org/abs/0902.3656
Author(s): Iosif Pinelis
Abstract: The well-known Bennett-Hoeffding bound for sums of independent random
variables is refined, by taking into account truncated third moments, and at
that also improved by using, instead of the class of all increasing exponential
functions, the much larger class of all generalized moment functions f such
that f and f" are increasing and convex. It is shown that the resulting bounds
have certain optimality properties. Comparisons with related known bounds are
given. The results can be extended in a standard manner to (the maximal
functions of) (super)martingales.
http://arxiv.org/abs/0902.4058
Author(s): Iosif Pinelis
Abstract: Integral expressions for positive-part moments E X_+^p (p>0) of random
variables X are presented, in terms of the Fourier-Laplace or Fourier
transforms of the distribution of X. A necessary and sufficient condition for
the validity of such an expression is given. This study was motivated by
extremal problems in probability and statistics, where one needs to evaluate
such positive-part moments.
http://arxiv.org/abs/0902.4214
Author(s): L. Vostrikova
Abstract: We study the differentiability of Bessel flow $\rho : x \to \rho ^x_t$, where
$(\rho ^x_t)_{t\geq 0}$ is BES $^x(\delta $) process of dimension $\delta >1$
starting from $x$. For $\delta \geq 2$ we prove the existence of bicontinuous
derivatives in P-a.s. sense at $x\geq 0$ and we study the asymptotic behaviour
of the derivatives at $x=0$. For $1< \delta <2$ we prove the existence of a
modification of Bessel flow having derivatives in probability sense at $x\geq
0$. We study the asymptotic behaviour of the derivatives at $t=\tau_0(x)$ where
$\tau_0(x)$ is the first zero of $(\rho ^x_t)_{t\geq 0}$.
http://arxiv.org/abs/0902.4232
Author(s): Sebastian del Ba\~no Rollin and Joan-Andreu L\'azaro-Cam\'i
Abstract: We introduce the concept of multidimensional antithetic as the absolute
minimum of the covariance function $O(N)\to\mathbb{R}$ defined by $A\mapsto
Cov(f(\xi),f(A\xi))$ where $\xi$ is a standard $N$-dimensional normal random
variable and $f:\mathbb{R}^{N}\to\mathbb{R}$ is an almost everywhere
differentiable function. The antithetic matrix is designed to optimise the
calculation of $E[f(\xi)]$ in a Monte Carlo simulation. We present an iterative
annealing algorithm that dynamically incorporates the estimation of the
antithetic matrix within the Monte Carlo calculation.
http://arxiv.org/abs/0902.4211
Author(s): Charles Bordenave (IMT) and Giovanni Luca Torrisi
Abstract: We analyze the asymptotic properties of an Euclidean optimization problem on
the plane. Specifically, we consider a network with 3 bins and n objects
spatially uniformly distributed, each object being allocated to a bin at a cost
depending on its position. Two allocations are considered: the allocation
minimizing the bin loads and the allocation allocating each object to its less
costly bin. We analyze the asymptotic properties of these allocations as the
number of objects grows to infinity. Using the symmetries of the problem, we
derive a law of large numbers, a central limit theorem and a large deviation
principle for both loads with explicit expressions. In particular, we prove
that the two allocations satisfy the same law of large numbers, but they do not
have the same asymptotic fluctuations and rate functions.
http://arxiv.org/abs/0902.4304
Author(s): V. Beffara and S. Friedli and Y. Velenik
Abstract: We describe the scaling limit of the nearest neighbour prudent walk on the
square lattice, which performs steps uniformly in directions in which it does
not see sites already visited. We show that the process eventually settles in
one of the quadrants, and derive its scaling limit, which can be expressed in
terms of a pair of independent stable subordinators. We also show that the
asymptotic speed of the walk is well-defined in the L_1 -norm and equals 3/7.
This process possesses unusual properties: it is ballistic but does not have an
asymptotic direction, and several natural observables display ageing.
http://arxiv.org/abs/0902.4312
Author(s): Michael R\"ockner and Yi Wang
Abstract: This note deals with existence and uniqueness of (variational) solutions to
the following type of stochastic partial differential equations on a Hilbert
space H dX(t) = A(t,X(t))dt + B(t,X(t))dW(t) + h(t) dG(t) where A and B are
random nonlinear operators satisfying monotonicity conditions and G is an
infinite dimensional Gaussian process adapted to the same filtration as the
cylindrical Wiener pocess W(t), t >= 0.
http://arxiv.org/abs/0902.4324
Author(s): Andreas E.Kyprianou and Xiaowen Zhou
Abstract: In the spirit of previous of Albrecher, Hipp, Renaud and Zhou we consider a
L\'evy insurance risk model with tax payments of a more general structure than
in the aforementioned papers that was also considered in \cite{ABBR}. In terms
of scale functions, we establish three fundamental identities of interest which
have stimulated a large volume of actuarial research in recent years. That is
to say, the two sided exit problem, the net present value of tax paid until
ruin as well as a generalized version of the Gerber-Shiu function. The method
we appeal to differs from former works in that we appeal predominantly to
excursion theory.
http://arxiv.org/abs/0902.4340
Author(s): E. Csaki and M. Csorgo and A. Foldes and P. Revesz
Abstract: We study the path behaviour of a simple random walk on the 2-dimensional comb
lattice ${\mathbb C}^2$ that is obtained from ${\mathbb Z}^2$ by removing all
horizontal edges off the x-axis. In particular, we prove a strong approximation
result for such a random walk which, in turn, enables us to establish strong
limit theorems, like the joint Strassen type law of the iterated logarithm of
its two components, as well as their marginal Hirsch type behaviour.
http://arxiv.org/abs/0902.4369
Author(s): T. S. Jackson and N. Read
Abstract: The minimum spanning tree (MST) is a combinatorial optimization problem:
given a connected graph with a real weight ("cost") on each edge, find the
spanning tree that minimizes the sum of the total cost of the occupied edges.
We consider the random MST, in which the edge costs are (quenched) independent
random variables. There is a strongly-disordered spin-glass model due to Newman
and Stein [Phys. Rev. Lett. 72, 2286 (1994)], which maps precisely onto the
random MST. We study scaling properties of random MSTs using a relation between
Kruskal's greedy algorithm for finding the MST, and bond percolation. We solve
the random MST problem on the Bethe lattice (BL) with appropriate wired
boundary conditions and calculate the fractal dimension D=6 of the connected
components. Viewed as a mean-field theory, the result implies that on a lattice
in Euclidean space of dimension d, there are of order W^{d-D} large connected
components of the random MST inside a window of size W, and that d = d_c = D =
6 is a critical dimension. This differs from the value 8 suggested by Newman
and Stein. We also critique the original argument for 8, and provide an
improved scaling argument that again yields d_c=6. The result implies that the
strongly-disordered spin-glass model has many ground states for d>6, and only
of order one below six. The results for MSTs also apply on the Poisson-weighted
infinite tree, which is a mean-field approach to the continuum model of MSTs in
Euclidean space, and is a limit of the BL. In a companion paper we develop an
epsilon=6-d expansion for the random MST on critical percolation clusters.
http://arxiv.org/abs/0902.3651
Author(s): Alessandra Faggionato and Davide Gabrielli and Marco Ribezzi Crivellari
Abstract: We consider a class of stochastic dynamical systems, called piecewise
deterministic Markov processes, with states $(x, \s)\in \O\times \G$, $\O$
being a region in $\bbR^d$ or the $d$--dimensional torus, $\G$ being a finite
set. The continuous variable $x$ follows a piecewise deterministic dynamics,
the discrete variable $\s$ evolves by a stochastic jump dynamics and the two
resulting evolutions are fully--coupled. We study stationarity, reversibility
and time--reversal symmetries of the process. Increasing the frequency of the
$\s$--jumps, we show that the system behaves asymptotically as deterministic
and we investigate the structure of fluctuations (i.e. deviations from the
asymptotic behavior), recovering in a non Markovian frame results obtained by
Bertini et al. \cite{BDGJL1, BDGJL2, BDGJL3, BDGJL4}, in the context of
Markovian stochastic interacting particle systems. Finally, we discuss a
Gallavotti--Cohen--type symmetry relation with involution map different from
time--reversal. For several examples the above results are recovered by
explicit computations.
http://arxiv.org/abs/0902.4195
Author(s): Guillaume Aubrun (ICJ)
Abstract: We present lower estimates for the best constant appearing in the weak (1,1)
maximal inequality in the space $(\R^n,\|\cdot\|_{\infty})$. We show that it
grows to infinity faster than $(\log n)^{\kappa}$ for any $\kappa <1$. We
follow the approach used by J.M. Aldaz in a recent paper. The new part of the
argument relies on Donsker's theorem identifying the Brownian bridge as the
limit $(n \to \infty)$ of the empirical distribution function associated to
coordinates of a point randomly chosen in the unit cube $[0,1]^n$.
http://arxiv.org/abs/0902.4305
Author(s): Zhenning Kong and Edmund M. Yeh
Abstract: We investigate the problem of disseminating broadcast messages in wireless
networks with time-varying links from a percolation-based perspective. Using a
model of wireless networks based on random geometric graphs with dynamic on-off
links, we show that the delay for disseminating broadcast information exhibits
two behavioral regimes, corresponding to the phase transition of the underlying
network connectivity. When the dynamic network is in the subcritical phase,
ignoring propagation delays, the delay scales linearly with the Euclidean
distance between the sender and the receiver. When the dynamic network is in
the supercritical phase, the delay scales sub-linearly with the distance.
Finally, we show that in the presence of a non-negligible propagation delay,
the delay for information dissemination scales linearly with the Euclidean
distance in both the subcritical and supercritical regimes, with the rates for
the linear scaling being different in the two regimes.
http://arxiv.org/abs/0902.4449
Author(s): V.I. Borzenko and Z.M. Lezina and A. K.Loginov and Ya.Yu. Tsodikova and and P.Yu. Chebotarev
Abstract: Consideration was given to a model of social dynamics controlled by
successive collective decisions based on the threshold majority procedures. The
current system state is characterized by the vector of participants' capitals
(utilities). At each step, the voters can either retain their status quo or
accept the proposal which is a vector of the algebraic increments in the
capitals of the participants. In this version of the model, the vector is
generated stochastically. Comparative utility of two social attitudes--egoism
and collectivism--was analyzed. It was established that, except for some
special cases, the collectivists have advantages, which makes realizable the
following scenario: on the conditions of protecting the corporate interests, a
group is created which is joined then by the egoists attracted by its
achievements. At that, group egoism approaches altruism. Additionally, one of
the considered variants of collectivism handicaps manipulation of voting by the
organizers.
http://arxiv.org/abs/0902.4460
Author(s): Martin Hairer and Jonathan C. Mattingly and Michael Scheutzow
Abstract: There are many Markov chains on infinite dimensional spaces whose one-step
transition kernels are mutually singular when starting from different initial
conditions. We give results which prove unique ergodicity under minimal
assumptions on one hand and the existence of a spectral gap under conditions
reminiscent of Harris' theorem. The first uses the existence of couplings which
draw the solutions together as time goes to infinity. Such "asymptotic
couplings" were central to recent work on SPDEs on which this work builds. The
emphasis here is on stochastic differential delay equations.Harris' celebrated
theorem states that if a Markov chain admits a Lyapunov function whose level
sets are "small" (in the sense that transition probabilities are uniformly
bounded from below), then it admits a unique invariant measure and transition
probabilities converge towards it at exponential speed. This convergence takes
place in a total variation norm, weighted by the Lyapunov function. A second
aim of this article is to replace the notion of a "small set" by the much
weaker notion of a "d-small set," which takes the topology of the underlying
space into account via a distance-like function d. With this notion at hand, we
prove an analogue to Harris' theorem, where the convergence takes place in a
Wasserstein-like distance weighted again by the Lyapunov function. This
abstract result is then applied to the framework of stochastic delay equations.
http://arxiv.org/abs/0902.4495
Author(s): Jonathan C. Mattingly and Scott A. McKinley and Natesh S. Pillai
Abstract: We consider a simple model for the fluctuating hydrodynamics of a flexible
polymer in dilute solution, demonstrating geometric ergodicity for a pair of
particles that interact with each other through a nonlinear spring potential
while being advected by a stochastic Stokes fluid velocity field. This is a
generalization of previous models which have used linear spring forces as well
as white-in-time fluid velocity fields.
We follow previous work combining control theoretic arguments, Lyapunov
functions, and hypo-elliptic diffusion theory to prove exponential convergence
via a Harris chain argument. To this, we add the possibility of excluding
certain "bad" sets in phase space in which the assumptions are violated but
from which the systems leaves with a controllable probability. This allows for
the treatment of singular drifts, such as those derived from the Lennard-Jones
potential, which is an novel feature of this work.
http://arxiv.org/abs/0902.4496
Author(s): Vijay G. Subramanian and Tara Javidi and Somsak Kittipiyakul
Abstract: In this paper, a many-sources large deviations principle (LDP) for the
transient workload of a multi-queue single-server system is established where
the service rates are chosen from a compact, convex and coordinate-convex rate
region and where the service discipline is the max-weight policy. Under the
assumption that the arrival processes satisfy a many-sources LDP, this is
accomplished by employing Garcia's extended contraction principle that is
applicable to quasi-continuous mappings.
For the simplex rate-region, an LDP for the stationary workload is also
established under the additional requirements that the scheduling policy be
work-conserving and that the arrival processes satisfy certain mixing
conditions.
The LDP results can be used to calculate asymptotic buffer overflow
probabilities accounting for the multiplexing gain, when the arrival process is
an average of \emph{i.i.d.} processes. The rate function for the stationary
workload is expressed in term of the rate functions of the finite-horizon
workloads when the arrival processes have \emph{i.i.d.} increments.
http://arxiv.org/abs/0902.4569
Author(s): Jean-Fran\c{c}ois Marckert (LaBRI) and Gr\'egory Miermont (ENS)
Abstract: We prove that a uniform, rooted unordered binary tree with $n$ vertices has
the Brownian continuum random tree as its scaling limit for the
Gromov-Hausdorff topology. The limit is thus, up to a constant factor, the same
as that of uniform plane trees or labeled trees. Our analysis rests on a
combinatorial and probabilistic study of appropriate trimming procedures of
trees.
http://arxiv.org/abs/0902.4570
Author(s): Robert C. Dalang and Marta Sanz-Sol\'e
Abstract: We develop several results on hitting probabilities of random fields which
highlight the role of the dimension of the parameter space. This yields upper
and lower bounds in terms of Hausdorff measure and Bessel-Riesz capacity,
respectively. We apply these results to a system of stochastic wave equations
in spatial dimension $k\ge1$ driven by a $d$-dimensional spatially homogeneous
additive Gaussian noise that is white in time and colored in space.
http://arxiv.org/abs/0902.4583
Author(s): Pavel Chebotarev
Abstract: In the simplest version of the model of group decision making in the
stochastic environment, the participants are segregated into egoists and a
group of collectivists. A "proposal of the environment" is a stochastically
generated vector of algebraic increments of capitals. The social dynamics is
determined by the sequence of proposals accepted by a majority voting (with a
threshold) of the participants. In this paper, we obtain analytical expressions
for the expected values of capitals for all the participants, including
collectivists and egoists. In addition, distinctions between some principles of
group voting are discussed.
http://arxiv.org/abs/0902.4514
Author(s): Sourav Chatterjee and Soumik Pal
Abstract: We consider a particular class of n-dimensional homogeneous diffusions all of
which have an identity diffusion matrix and a drift function that is piecewise
constant and scale invariant. Abstract stochastic calculus immediately gives us
general results about existence and uniqueness in law and invariant probability
distributions when they exist. These invariant distributions are probability
measures on the $n$-dimensional space and can be extremely resistant to a more
detailed understanding. To have a better analysis, we construct a polyhedra
such that the inward normal at its surface is given by the drift function and
show that the finer structures of the invariant probability measure is
intertwined with the geometry of the polyhedra. We show that several natural
interacting Brownian particle models can thus be analyzed by studying the
combinatorial fan generated by the drift function, particularly when these are
simplicial. This is the case when the polyhedra is a polytope that is invariant
under a Coxeter group action, which leads to an explicit description of the
invariant measures in terms of iid Exponential random variables. Another class
of examples is furnished by interactions indexed by weighted graphs all of
which generate simplicial polytopes with $n !$ faces. We show that the
proportion of volume contained in each component simplex corresponds to a
probability distribution on the group of permutations, some of which have
surprising connections with the classical urn models.
http://arxiv.org/abs/0902.4762
Author(s): Andrei Y. Yakovlev and Nikolay M. Yanev
Abstract: This paper considers the relative frequencies of distinct types of
individuals in multitype branching processes. We prove that the frequencies are
asymptotically multivariate normal when the initial number of ancestors is
large and the time of observation is fixed. The result is valid for any
branching process with a finite number of types; the only assumption required
is that of independent individual evolutions. The problem under consideration
is motivated by applications in the area of cell biology. Specifically, the
reported limiting results are of advantage in cell kinetics studies where the
relative frequencies but not the absolute cell counts are accessible to
measurement. Relevant statistical applications are discussed in the context of
asymptotic maximum likelihood inference for multitype branching processes.
http://arxiv.org/abs/0902.4773
Author(s): Rick Durrett and Lea Popovic
Abstract: We consider two processes that have been used to study gene duplication,
Watterson's [Genetics 105 (1983) 745--766] double recessive null model and
Lynch and Force's [Genetics 154 (2000) 459--473] subfunctionalization model.
Though the state spaces of these diffusions are two and six-dimensional,
respectively, we show in each case that the diffusion stays close to a curve.
Using ideas of Katzenberger [Ann. Probab. 19 (1991) 1587--1628] we show that
one-dimensional projections converge to diffusion processes, and we obtain
asymptotics for the time to loss of one gene copy. As a corollary we find that
the probability of subfunctionalization decreases exponentially fast as the
population size increases. This rigorously confirms a result Ward and Durrett
[Theor. Pop. Biol. 66 (2004) 93--100] found by simulation that the likelihood
of subfunctionalization for gene duplicates decays exponentially fast as the
population size increases.
http://arxiv.org/abs/0902.4780
Author(s): Boris Buchmann and Ngai Hang Chan
Abstract: Consider $Z^f_t(u)=\int_0^{tu}f(N_s) ds$, $t>0$, $u\in[0,1]$, where
$N=(N_t)_{t\in\mathbb{R}}$ is a normal process and $f$ is a measurable
real-valued function satisfying $Ef(N_0)^2<\infty$ and $Ef(N_0)=0$. If the
dependence is sufficiently weak Hariz [J. Multivariate Anal. 80 (2002)
191--216] showed that $Z_t^f/t^{1/2}$ converges in distribution to a multiple
of standard Brownian motion as $t\to\infty$. If the dependence is sufficiently
strong, then $Z_t/(EZ_t(1)^2)^{1/2}$ converges in distribution to a higher
order Hermite process as $t\to\infty$ by a result by Taqqu [Wahrsch. Verw.
Gebiete 50 (1979) 53--83]. When passing from weak to strong dependence, a
unique situation encompassed by neither results is encountered. In this paper,
we investigate this situation in detail and show that the limiting process is
still a Brownian motion, but a nonstandard norming is required. We apply our
result to some functionals of fractional Brownian motion which arise in time
series. For all Hurst indices $H\in(0,1)$, we give their limiting
distributions. In this context, we show that the known results are only
applicable to $H<3/4$ and $H>3/4$, respectively, whereas our result covers
$H=3/4$.
http://arxiv.org/abs/0902.4784
Author(s): S. N. Lahiri and S. Sun
Abstract: This paper proves a Berry--Esseen theorem for sample quantiles of
strongly-mixing random variables under a polynomial mixing rate. The rate of
normal approximation is shown to be $O(n^{-1/2})$ as $n\to\infty$, where $n$
denotes the sample size. This result is in sharp contrast to the case of the
sample mean of strongly-mixing random variables where the rate $O(n^{-1/2})$ is
not known even under an exponential strong mixing rate. The main result of the
paper has applications in finance and econometrics as financial time series
data often are heavy-tailed and quantile based methods play an important role
in various problems in finance, including hedging and risk management.
http://arxiv.org/abs/0902.4796
Author(s): Mark Craddock and Kelly A. Lennox
Abstract: This paper uses Lie symmetry methods to calculate certain expectations for a
large class of It\^{o} diffusions. We show that if the problem has sufficient
symmetry, then the problem of computing functionals of the form
$E_x(e^{-\lambda X_t-\int_0^tg(X_s) ds})$ can be reduced to evaluating a single
integral of known functions. Given a drift $f$ we determine the functions $g$
for which the corresponding functional can be calculated by symmetry.
Conversely, given $g$, we can determine precisely those drifts $f$ for which
the transition density and the functional may be computed by symmetry. Many
examples are presented to illustrate the method.
http://arxiv.org/abs/0902.4806
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