[PAS] Probability Abstracts 108
Probability Abstract Service
pas at lists.imstat.org
Tue Mar 3 02:14:14 CST 2009
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.
This letter can be also found on line at
http://pas.imstat.org/Letters/letter_108.shtml
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7954. Regularity of Ornstein-Uhlenbeck processes driven by a L{\'e}vy
white noise
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
7955. Weak Solutions of the Stochastic Landau-Lifshitz-Gilbert Equation
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
7956. Conditions for certain ruin for the generalised Ornstein-
Uhlenbeck process and the structure of the upper and lower bounds
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
7957. Current and density fluctuations for interacting particle
systems with anomalous diffusive behavior
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
7958. Order-invariant Measures on Causal Sets
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
7959. Spatial Epidemics and Local Times for Critical Branching Random
Walks in Dimensions 2 and 3
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
7960. Representation of gaussian small ball probabilities in $l_2$
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
7961. Adjustment coefficient for risk processes in some dependent
contexts
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
7962. Maximum Entropy on Compact Groups
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
7963. p-Adic Spherical Coordinates and Their Applications
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
7964. Order-invariant Measures on Fixed Causal Sets
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
7965. Beta Jacobi processes
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
7966. Stein's lemma, Malliavin calculus, and tail bounds, with
application to polymer fluctuation exponent
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
7967. General discrete random walk with variable absorbing probabilities
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
7968. Random Current Representation for Transverse Field Ising Models
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
7969. Invariant manifolds for random and stochastic partial
differential equations
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
7970. An upper bound for front propagation velocities inside moving
populations
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
7971. A unifying formulation of the Fokker-Planck-Kolmogorov equation
for general stochastic hybrid systems (extended version)
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
7972. Infinite rate mutually catalytic branching in infinitely many
colonies. Construction, characterization and convergence
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
7973. On the growth of the supercritical long-range percolation
cluster on $\mathbb{Z}^d$ and an application for spatial epidemics
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 $1 < a_2 <
\infty$ such that $\lim_{k\to \infty} \mathbb{P}(a_1<|\mathcal{B}_k|
^{1/k}< a_2) = 1$, $|\mathcal{B}_k|^{1/k} \to 1$ almost surely. This
result can be applied to spatial $SIR$ epidemics. In particular, we
show that it is possible to construct a distribution of long-range
contacts between individuals only depending on their distance, such
that the number of infectious individuals in the $k$-th infection
generation stochastically dominates an exponentially growing function.
http://arxiv.org/abs/0901.0661
7974. Isomorphism and Symmetries in Random Phylogenetic Trees
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
7975. Convolution symmetries of integrable hierarchies, matrix models
and $\tau$-functions
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
7976. New bounds for the free energy of directed polymer in dimension
1+1 and 1+2
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
7977. Phantom Probability
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
7978. A new approach to mutual information. II
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
7979. When do nonlinear filters achieve maximal accuracy?
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
7980. A CLT for the L^{2} modulus of continuity of Brownian local time
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
7981. Asymptotic behaviour of a general reversible chemical reaction-
diffusion equation
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
7982. Projecting the Fokker-Planck Equation onto a finite dimensional
exponential family
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
7983. Collisions and Spirals of Loewner Traces
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
7984. A Better Way to Deal the Cards
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
7985. Semi-infinite TASEP with a Complex Boundary Mechanism
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
7986. Two kinds of conditionings for stable L\'evy processes
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
7987. Mixture of the Riesz distribution with respect to the
multivariate Poisson
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
7988. Tails of multivariate Archimedean copulas
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
7989. The phase transition of the quantum Ising model is sharp
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
7990. A cautionary tale on the efficiency of some adaptive Monte Carlo
Schemes
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
7991. The Logarithmic Sobolev Inequality in Infinite dimensions for
Unbounded Spin Systems on the Lattice with non Quadratic Interactions
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
7992. Degree-distribution Stability of Evolving Networks
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
7993. Perturbing the Logarithmic Sobolev Inequality for Unbounded Spin
Systems on the Lattice with non Quadratic Interactions
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
7994. Correlation inequalities of GKS type for the Potts model
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
7995. Entropic Measure on Multidimensional Spaces
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
7996. Approximation of target problems in Blackwell spaces
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
7997. Distribution of Random Variables on the Symmetric Group
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
7998. Classification of E_0--Semigroups by Product Systems
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
7999. A finite dimensional filter with exponential conditional density
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
8000. Brownian motion with respect to time-changing Riemannian
metrics, applications to Ricci flow
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
8001. Some differential systems driven by a fBm with Hurst parameter
greater than 1/4
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
8002. The cut metric, random graphs, and branching processes
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
8003. Hankel determinants of Dirichlet series
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
8004. A Lower Bound on the Capacity of Wireless Erasure Networks with
Random Node Locations
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
8005. Soliton dynamics for the Korteweg-de Vries equation with
multiplicative homogeneous noise
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
8006. H"older index for density states of (alpha,1,beta)-
superprocesses at a given point
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
8007. On weak approximation of U-statistics
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
8008. An Excursion-Theoretic Approach to Stability of Discrete-Time
Stochastic Hybrid Systems
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
8009. Counterexamples in the theory of fair division
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
8010. Pricing and trading credit default swaps in a hazard process model
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
8011. Poisson process approximation for dependent superposition of
point processes
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
8012. Busemann functions and equilibrium measures in last passage
percolation
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
8013. Asymptotic optimality of maximum pressure policies in stochastic
processing networks
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
8014. State-dependent Foster-Lyapunov criteria for subgeometric
convergence of Markov chains
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
8015. Central limit theorem for the solution of the Kac equation
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
8016. The asymptotic distribution and Berry--Esseen bound of a new
test for independence in high dimension with an application to
stochastic optimization
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
8017. Optimal stopping and free boundary characterizations for some
Brownian control problems
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
8018. The contact process in a dynamic random environment
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
8019. A von Neumann theorem for uniformly distributed sequences of
partitions
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
8020. Fermionic construction of partition functions for two-matrix
models and perturbative Schur function expansions
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
8021. Synchronization of dissipative dynamical systems driven by non-
Gaussian Levy noises
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
8022. Exact Asymptotic for the Tail of Maximum of Smooth Random Field
Distribution
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
8023. Averaging of Hamiltonian flows with an ergodic component
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
8024. Optimal approximation rate of certain stochastic integrals
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
8025. Weak solutions for forward--backward SDEs--a martingale problem
approach
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
8026. Some local approximations of Dawson--Watanabe superprocesses
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
8027. Trivial intersection of $\sigma$-fields and Gibbs sampling
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
8028. Ornstein-Uhlenbeck Equations with time-dependent coefficients
and Levy Noise in finite and infinite dimensions
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
8029. Evolution Systems of Measures for Non-autonomous Ornstein-
Uhlenbeck Processes with Levy noise
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
8030. Depinning of a polymer in a multi-interface medium
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
8031. A martingale approach to continuous time marginal structural
models
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
8032. The compositional construction of Markov processes
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
8033. Ham Sandwich with Mayo: A Stronger Conclusion to the Classical
Ham Sandwich Theorem
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
8034. A Trotter type approach to infinite rate mutually catalytic
branching
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
8035. Condenser physics applied to Markov chains - A brief
introduction to potential theory
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
8036. Simulation and approximation of Levy-driven stochastic
differential equations
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
8037. On the Convergence of the Ensemble Kalman Filter
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
8038. A process very similar to multifractional Brownian motion
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
8039. Max-plus Stochastic Control and Risk-sensitivity
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
8040. Factorization of Joint Probability Mass Functions into Parity
Check Interactions
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
8041. Moderate deviations in random graphs and Bernoulli random matrices
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
8042. From the long jump random walk to the fractional Laplacian
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
8043. Limit theorems for random spatial drainage networks
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
8044. The algebraic difference of two random Cantor sets: the Larsson
family
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
8045. A Stochastic Approach for Parameterizing Unresolved Scales in a
System with Memory
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
8046. The mean width of circumscribed random polytopes
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
8047. The Asymptotic Shape Theorem for Generalized First Passage
Percolation
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
8048. Excursions of the integral of the Brownian motion
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
8049. Expansion of the propagation of chaos for Bird and Nanbu systems
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
8050. Normal approximation for isolated balls in an urn allocation model
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
8051. Zonal polynomials and hypergeometric functions of quaternion
matrix argument
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
8052. The mean width of random polytopes circumscribed around a convex
body
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
8053. Generalized Whittle-Mat$\acute{\text{E}}$rn random field as a
model of correlated fluctuations
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
8054. Logconcave Random Graphs
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
8055. On a random number of disorders
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
8056. On the global maximum of the solution to a stochastic heat
equation with compact-support initial data
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
8057. A phase diagram for a stochastic reaction diffusion system
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
8058. New Classes of Infinitely Divisible Distributions Related to the
Goldie-Steutel-Bondesson Class
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
8059. Affine Diffusion Processes: Theory and Applications
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
8060. Limiting behaviors of the Brownian motions on hyperbolic spaces
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
8061. Growth Rates and Explosions in Sandpiles
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
8062. Generalized kinetic Maxwell type models of granular gases
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
8063. Choice-memory tradeoff in allocations
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
8064. Heat kernel estimates and Harnack inequalities for some
Dirichlet forms with non-local part
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
8065. On the Limiting Shape of Random Young Tableaux Associated to
Inhomogeneous Words
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
8066. Mixing time of critical Ising model on trees is polynomial in
the height
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
8067. Discretization-invariant Bayesian inversion and Besov space priors
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
8068. Note: Random-to-front shuffles on trees
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
8069. Excited against the tide: A random walk with competing drifts
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
8070. Uniform shrinking and expansion under isotropic Brownian flows
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
8071. Regeneration in Random Combinatorial Structures
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
8072. Exact confidence intervals for the Hurst parameter of a
fractional Brownian motion
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
8073. Universality of the Pearcey process
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
8074. Is the critical percolation probability local?
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
8075. Nagaev method via Keller-Liverani theorem
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
8076. A survey on dynamical percolation
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
8077. A stochastic calculus for multidimensional fractional Brownian
motion with arbitrary Hurst index
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
8078. Weak KAM methods and ergodic optimal problems for countable
Markov shifts
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
8079. Ergodicity of multiplicative statistics
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
8080. Scaled limit and rate of convergence for the largest eigenvalue
from the generalized Cauchy random matrix ensemble
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
8081. Wick Calculus For Nonlinear Gaussian Functionals
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
8082. Parameter estimation for fractional Ornstein-Uhlenbeck processes
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
8083. A note on adiabatic theorem for Markov chains and adiabatic
quantum computation
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
8084. On generalized Cauchy-Stieltjes transforms of some Beta
distributions
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
8085. On Brownian motion on the plane with membranes on rays with a
common endpoint
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
8086. Palm pairs and the general mass-transport principle
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
8087. Cutpoints and resistance of random walk paths
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
8088. A passage to the Poisson-Dirichlet through the Bessel square
processes
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
8089. Variance decay for functionals of the environment viewed by the
particle
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
8090. Critical behavior in inhomogeneous random graphs
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
8091. Shelf Life of Candidates in the Generalized Secretary Problem
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
8092. On Stein's method for multivariate normal approximation
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
8093. Fermionic construction of tau functions and random processes
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
8094. Clustering Bounds on N-Point Correlations for Unbounded Spin
Systems
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
8095. Very large graphs
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
8096. Carries, shuffling, and symmetric functions
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
8097. Poset limits and exchangeable random posets
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
8098. Random symmetrizations of measurable sets
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
8099. A L\'{e}vy input model with additional state-dependent services
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
8100. Discretizing the fractional Levy area
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
8101. Convergence of multi-class systems of fixed possibly infinite
sizes
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
8102. A Bernstein type inequality and moderate deviations for weakly
dependent sequences
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
8103. Variance limite d'une marche al\'eatoire r\'eversible en milieu
al\'eatoire sur Z (Limit of the Variance of a Reversible Random Walk
in Random Medium on Z)
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
8104. Belief propagation : an asymptotically optimal algorithm for the
random assignment problem
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
8105. Heat Conduction Networks: Disposition of Heat Baths and
Invariant Measure
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
8106. On Small Perturbations of a Spin Glass System
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
8107. Some Rigorous Results on Semiflexible Polymers. I. Free and
confined polymers
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
8108. A Finitization of the Bead Process
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
8109. A few ideas about quantitative convergence of collison models to
the mean field limit
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
8110. Isoperimetry for spherically symmetric log-concave probability
measures
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
8111. Correlated Drainage Model
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
8112. Total Current Fluctuations in ASEP
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
8113. Univariate approximations in the infinite occupancy scheme
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
8114. Translated Poisson approximation to equilibrium distributions of
Markov population processes
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
8115. Local limit approximations for Markov population processes
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
8116. Random Walks on Directed Covers of Graphs
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
8117. About Gaussian filtering problems with general exponential
quadratic criteria
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
8118. Randomized Kaczmarz solver for noisy linear systems
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
8119. Transience/Recurrence and the speed of a one-dimensional random
walk in a "have your cookie and eat it" environment
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
8120. Multiple orthogonal polynomial ensembles
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
8121. Extremes of Levy processes with light tails
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
8122. Asymptotic directions in random walks in random environment
revisited
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
8123. Probabilistic Representation of Weak Solutions of Partial
Differential Equations with Polynomial Growth Coefficients
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
8124. On the spread of supercritical random graphs
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
8125. Asymptotic Expansions for the Sojourn Time Distribution in the
$M/G/1$-PS Queue
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
8126. On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the
Number of Other Users
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
8127. Uniform bounds for exponential moments of maximum of Dyck paths
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
8128. On Azema-Yor processes, their optimal properties and the
Bachelier-Drawdown equation
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
8129. The martingale problem for Markov solutions to the Navier-Stokes
equations
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
8130. An almost sure energy inequality for Markov solutions to the 3D
Navier-Stokes equations
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
8131. Random Graphons and a Weak Positivstellensatz for Graphs
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
8132. Bilinear and Quadratic Variants on the Littlewood-Offord Problem
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
8133. Homogenization of locally stationary diffusions with possibly
degenerate diffusion matrix
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
8134. A simple construction of Werner measure from chordal SLE$_{8/3}$
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
8135. Gaussian density estimates for solutions to quasi-linear
stochastic partial differential equations
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
8136. The critical Z-invariant Ising model via dimers: locality property
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
8137. Randomness on Computable Probability Spaces - A Dynamical Point
of View
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
8138. Cover Time and Broadcast Time
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
8139. Mesoscopic fluctuations of the zeta zeros
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
8140. On the diameter of the set of satisfying assignments in random
satisfiable k-CNF formulas
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
8141. Batch queues, reversibility and first-passage percolation
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
8142. Transportation-information inequalities for Markov processes
(II) : relations with other functional inequalities
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
8143. A Single Server Retrial Queue with Different Types of Server
Interruptions
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
8144. Burkholder-Davis-Gundy type Inequalities of the It\^o stochastic
integral with respect to Levy noise on Banach spaces
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
8145. Stochastic approach for the subordination in Bochner sense
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
8146. A new look at the Heston characteristic function
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
8147. Heavy-traffic analysis of the maximum of an asymptotically
stable random walk
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
8148. M/M/1 Queueing System with Non-preemptive Priority
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
8149. Distribution-valued heavy-traffic limits for the $G/GI/\infty$
queue
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
8150. A note on the Poisson boundary of lamplighter random walks
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
8151. Limit theorems for Parrondo's paradox
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
8152. The determinacy of infinite games with eventual perfect monitoring
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
8153. Some results on random circulant matrices
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
8154. Heat kernel analysis on semi-infinite Lie groups
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
8155. Expansions for Gaussian processes and Parseval frames
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
8156. Integral Equations and the First Passage Time of Brownian Motions
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
8157. The Policy Iteration Algorithm for Average Continuous Control of
Piecewise Deterministic Markov Processes
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
8158. Li-Yau Type Gradient Estimates and Harnack Inequalities by
Stochastic Analysis
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
8159. Existence of an Optimal Control for Stochastic Systems with
Nonlinear Cost Functional
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
8160. Regularity of the Optimal Stopping Problem for Levy Processes
with Non-Degenerate Diffusions
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
8161. A Simulation Approach to Optimal Stopping Under Partial
Information
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
8162. A presentation of the category of stochastic matrices
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
8163. Random Walks in the Quarter Plane Absorbed at the Boundary :
Exact and Asymptotic
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
8164. Continuous Model for Homopolymers
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
8165. Fractional multiplicative processes
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
8166. Random repeated quantum interactions and random invariant states
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
8167. Bounds on the Location of the Maximum Stirling Numbers of the
Second Kind
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
8168. Irreducibility and uniqueness of stationary distribution
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
8169. Backward SDEs with superquadratic growth
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
8170. Quenched scaling limits of trap models
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
8171. Anisotropic Young diagrams and infinite-dimensional diffusion
processes with the Jack parameter
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
8172. Effect of Noise on Front Propagation in Reaction-Diffusion
equations of KPP type
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
8173. Finitely-additive measures on the asymptotic foliations of a
Markov compactum
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
8174. A (rough) pathwise approach to fully non-linear stochastic
partial differential equations
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
8175. Periodic homogenization with an interface: the one-dimensional
case
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
8176. Interacting Brownian motions in infinite dimensions with
logarithmic interaction potentials
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
8177. Large Deviations and Moments for the Euler Characteristic of a
Random Surface
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
8178. Single-crossover dynamics: finite versus infinite populations
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
8179. A better algorithm for random k-SAT
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
8180. On the re-rooting invariance property of Levy trees
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
8181. Thick Points of the Gaussian Free Field
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
8182. Asymptotic Independence of the Extreme Eigenvalues of GUE
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
8183. Equilibrium Fluctuations for the Totally Asymmetric Zero Range
process
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
8184. Rare event simulation for T-cell activation
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
8185. Levy flights and Levy -Schroedinger semigroups
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
8186. Space-time covariance functions with compact support
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
8187. On the Bennett-Hoeffding inequality
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
8188. Positive-part moments via the Fourier-Laplace transform
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
8189. On regularity properties of Bessel flow
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
8190. Antithetic variates in higher dimensions
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
8191. Load optimization in a planar network
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
8192. Scaling Limit of the Prudent Walk
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
8193. A Note on variational solutions to SPDE perturbed by Gaussian
noise in a general class
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
8194. General tax structures and the Levy insurance risk model
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
8195. Strong limit theorems for a simple random walk on the 2-
dimensional comb
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
8196. Theory of minimum spanning trees I: Mean-field theory and
strongly disordered spin-glass model
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
8197. Stationarity, time--reversal and fluctuation theory for a class
of piecewise deterministic Markov processes
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
8198. Maximal inequality for high-dimensional cubes: quantitative
estimates
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
8199. Connectivity, Percolation, and Information Dissemination in
Large-Scale Wireless Networks with Dynamic Links
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
8200. Strategies of Voting in Stochastic Environment: Egoism and
Collectivism
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
8201. Asymptotic coupling and a weak form of Harris' theorem with
applications to stochastic delay equations
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
8202. Geometric ergodicity of a bead-spring pair with stochastic
Stokes forcing
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
8203. Many-Sources Large Deviations for Max-Weight Scheduling
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
8204. The CRT is the scaling limit of unordered binary trees
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
8205. Criteria for hitting probabilities with applications to systems
of stochastic wave equations
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
8206. Analytical Expression of the Expected Values of Capital at
Voting in the Stochastic Environment
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
8207. A combinatorial analysis of interacting diffusions
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
8208. Relative frequencies in multitype branching processes
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
8209. Degenerate diffusions arising from gene duplication models
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
8210. Integrated functionals of normal and fractional processes
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
8211. A Berry--Esseen theorem for sample quantiles under weak dependence
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
8212. The calculation of expectations for classes of diffusion
processes by Lie symmetry methods
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
-----------------------------------
Stefano M. Iacus
Department of Economics,
Business and Statistics
University of Milan
Via Conservatorio, 7
I-20123 Milan - Italy
Ph.: +39 02 50321 461
Fax: +39 02 50321 505
http://www.economia.unimi.it/iacus
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