1999

1999
Volume 5
Issue 1

M. Löwe
The Storage Capacity of Generalized Hopfield Models with Semantically Correlated Patterns
 pp. 1-19
     We analyze the storage capacity of a variant of the Hopfield model with semantically
correlated patterns $\xi_i^{\nu}$ (that is the patterns $\xi^\nu$ themselves are correlated
but consist of independent components $\xi_{i}^{\nu}$). We show that there is a class
of generalized Hopfield models of neural networks with $N$ neurons that can store
$N/{(\gamma \log N)}$ or $\alpha N$ spatially correlated patterns (depending on which
notion of storage is used), provided that the correlation comes from a homogeneous Markov
chain. The quantities $\gamma$ and $\alpha$ are independent of the correlations such that
these generalized Hopfield models may be regarded as the legitimate representative of the
standard Hopfield model in the presence of semantically correlated data.
     Keywords: Hopfield model, neural networks, storage capacity, Markov chains, large
deviations.

S. Sellami 
Equilibrium Density Fluctuations of a One-Dimensional Non-Gradient Reversible Model:
The Generalized Exclusion Process
 pp. 21-51
      We study the equilibrium density fluctuation fields of a one-dimensional reversible model.
We prove, for the generalized exclusion process, the Boltzmann - Gibbs principle.  This
principle, first introduced by Brox and Rost, is the basic stage which enables us to show
afterwards that our process converges in distribution to a generalized Ornstein - Uhlenbeck
process, by applying Holley and Stroock's theory.
      Keywords: fluctuations, Boltzmann - Gibbs principle, exclusion process, hydrodynamic
limits.

N.E. Ratanov 
Telegraph Evolutions in Inhomogeneous Media
 pp. 53-68
      We consider a random walk on a line with abrupt changes in velocity directions at
Poisson times. It is known due to M. Kac (1959) that the probability distribution of this
motion fits in the so-called telegraph equation. We generalize this observation to the case
of motions in inhomogeneous and in anisotropic media. The probabilities related to such
motions with reflectors and traps are calculated as the solutions of boundary value problems
for telegraph equation.
      Keywords: random evolutions, Poisson process, telegraph equation, diffusion
approximation.

A. Ambroladze, H. Wallin
Random Iteration of Isometries of the Hyperbolic Plane
 pp. 69-88
      Suppose that we start at a point $Z_0\in\overline{\C}$ in the extended complex plane
and form an orbit $\{Z_n\}^\infty_0$ by random iteration in the following way. At each step
$n\ge1$ we form $Z_n=f(Z_{n-1})$ where $f=f_n$ is chosen, with equal probability, as one
of the functions $-5/(1+z)$ and $-0.5/(1+z)$ which map the upper half-plane onto itself
preserving the hyperbolic metric. In [A. Barrlund, H. Wallin and J. Karlsson, Iteration of
Möbius transformations and attractors on the real line, Comput. Math. Appl., 1997, 33,
1-12] it was proved that the orbit $\{Z_n\}^\infty_0$ tends to the extended real line
$\overline{R}=R\cup\{\infty\}$ with probability 1, as $n$ tends to infinity. In this paper
we study the same problem for general isometries (in the hyperbolic metric) of the upper
half-plane onto itself. We prove that $Z_n \to \overline{R}$ in probability, as $n \to \infty$
and show that we do not have convergence with probability one in general. As an application
we study random iteration of linear maps on $R^2$ with determinant one. The problems studied
here are related to Markov chains and to the generation of fractal pictures as attractors
of iterated function systems.
     Keywords: random iteration, probabilistic attractor, hyperbolic isometry, Möbius
transformation, linear map.

F. Delcoigne,  G. Fayolle
Thermodynamical Limit and Propagation of Chaos in Polling Systems
 pp. 89-124
     We consider a sequence $\{P^{(N)}, N \geq 1 \}$ of standard polling networks,
consisting of $N$ nodes attended by $V^{(N)}$ mobile servers. When a server arrives at a
node $i$, he serves one of the waiting customers, if any, and then moves to node $j$ with
probability $p_{ij}^{(N)}$. Customers arrive according to a Poisson process. Service
requirements and switch-over times between nodes are independent exponentially distributed
random variables. The behaviour of $P^{(N)}$ is analyzed in thermodynamic limit, i.e. when
both $N$ and $V^{(N)}$ tend to infinity, with $U\egaldef\lim_{N\rightarrow\infty}V^{(N)}/N$
and $ 0 < U < \infty $. First, ergodicity conditions are given. Then, combining the
mean-field approximation approach together with weak convergence of Markov processes, the
joint distribution (customers, vehicles) for an arbitrary finite number of nodes is
explicitly characterized. In fact this distribution has a product form, which is the
mathematical analogue of the propagation of chaos. The speed of convergence is also computed.
In most of the study, $P^{(N)}$ is a fully symmetrical network, but a generalization is
carried out for systems provided with only blockwise symmetry.
     Keywords: chaos, network, polling, random walk, recurrence, transience,
thermodynamic limit.



1999
Volume 5
Issue 2

L. Bertini, B. Zegarlinski
Coercive Inequalities for Kawasaki Dynamics. The Product Case
 pp. 125-162
     We prove the Generalized Nash and Logarithmic Nash inequalities for a product measure
with Dirichlet form associated to the Kawasaki dynamics.
     Keywords: Generalized Nash, Logarithmic Nash inequalities, Bernoulli measures, Kawasaki
dynamics.

F. Paulin
Propriétés Asymptotiques des Relations d'Équivalences
Mesurées Discrètes
 pp. 163-200
     We study the asymptotic properties of orbits of a measurable action on a probability
space of a finitely generated group. We prove that if the measure in ergodic, stationary
and non-invariant, then almost every orbit is transient. We prove that if the measure is
stationary, then almost every orbit has 0,1,2 or a Cantor set of ends, and that almost every
orbit with two ends has linear growth.
     Keywords: measurable group actions, measured equivalence relations, ends, growth, random
walks.

N. Guillotin
Asymptotics of a Dynamic Random Walk in a Random Scenery: II. A Functional
Limit Theorem
 pp. 201-218
     Let $(S_{n})_{n\in{ N}}$ be a $Z$-random walk on nearest neighbours with dynamical
quasiperiodic transition probabilities in a random scenery $\xi(\alpha),\alpha\in Z$, that
is a family of i.i.d. random variables, independent of the random walk. It is shown that
$\displaystyle n^{-\frac{3}{4}}\sum_{i=0}^{[nt]} \xi(S_{i})$ converges weakly as $n\rightarrow
\infty$ to a self-similar process with stationary increments.
     Keywords: random walk, random scenery, local time, Brownian motion, functional limit
theorem.

E. Bertin, J.-M. Billiot, R. Drouilhet
k-Nearest-Neighbours Gibbs Processes
 pp. 219-234
     The present study introduces a class of directed $k$-nearest-neighbours point processes.
The neighbourhood relation is non-symmetric and depends on the realization of the process.
The existence of stationary Gibbs states is proved for such models in $R^m$. Furthermore,
under a finite range condition, uniqueness of stationary Gibbs states is proved for
sufficiently small intensity. Finally, original simulations are proposed using a direct
adaptation to our class of models of the Geyer and Møller algorithm.
     Keywords:
     stochastic geometry, Gibbs point processes, k-nearest-neighbours graph, equilibrium
equations, correlation functions.



1999
Volume 5
Issue 3

J.T. Lewis, C.-E. Pfister, W.G. Sullivan
Generic Points for Stationary Measures via
Large Deviation Theory
 pp. 235-267
     The construction of generic points by concatenation is discussed in the natural setting
of Large Deviation Theory. Our main result is the construction of generic points for any
stationary $k$-Markov measure $\alpha$, using only the $(k+1)$-marginals of $\alpha$ (Section 4).
This construction is based on the notion of LD-regular sequences and an improvement of results
in [J.T. Lewis, C.-E. Pfister and W.G. Sullivan, Entropy, concentration of probability and
conditional limit theorems. Markov Processes Relat. Fields, 1995, 1, 319-386]
about large deviations of conditioned measures (Section 6). The first part of the paper
provides motivation for the concatenation method coming from our recent study of Asymptotic
Equipartition Property [J.T. Lewis, C.-E. Pfister, R. Russel, W.G. Sullivan, Reconstruction
sequences and equipartition measures: an examination of the asymptotic equipartition property.
IEEE Inform. Theory, 1997, 43, 1935-1947].
     Keywords: large deviations, ergodic theory, entropy, generic points, normal numbers.

F. Comets, O. Zeitouni
Information Estimates and Markov Random Fields
 pp. 269-291
     Consider a random field, with values in some finite set $\Sigma\subset R$ and index
set a cube $\Lambda_n \subset Z^d$. We show that in the vicinity (in the information-
theoretic sense) of strongly mixing Markov fields, considering sub-blocks of variables
indexed by $\Lambda_m\subset \Lambda_n$, the distribution on this smaller cube can be
described precisely, even when the size of the cube grows with $n$. The general results
are then applied to mean field perturbations of Gibbs measures (in particular, mean field
perturbations of Ising models). The proofs use entropy arguments as well as (known) result
on complete analyticity and mixing for the Ising model.
     Keywords: information, relative entropy, Gibbs fields, Markov fields, weak-mixing,
complete analyticity, mean-field.

D. Crisan, P. Del Moral, T. Lyons
Discrete Filtering Using Branching and Interacting
Particle Systems
 pp. 293-318
      The stochastic filtering problem deals with the estimation of the current state of a
signal process given the information supplied by an associate process, usually called the
observation process. The aim of the current paper is to provide a unified and simple
approach for proving the validity of a series of numerical algorithms designed for solving
discrete time filtering problems. These algorithms appeared in various papers (see
[D. Crisan and T.J. Lyons, Nonlinear filtering and measure valued processes, Probab. Theory
and Relat. Fields, 1997, 109, 217-244; D. Crisan and T.J. Lyons, Convergence of a branching
particle method to the solution of the Zakai equation, SIAM J. Appl. Probab., 1998, 58 (5),
1568-1590; D. Crisan, J. Gaines and T.J. Lyons, A particle approximation of the solution
of the Kushner - Stratonovitch equation, To appear in Probab. Theory and Relat. Fields;
P. Del Moral, Non-linear filtering: interacting particle solution, Markov Processes Relat.
Fields, 1996, v.2, 555-580; P. Del Moral, Measure valued processes and interacting particle
systems. Application to non linear filtering problems, Publications du Laboratoire
de Statistiques et Probabilités, Université Paul Sabatier, 1996, 15-96,
To appear in Ann. Appl. Probab.; P. Del Moral, Filtrage non linéaire par systèmes
de particules en intéraction, C. R. Acad. Sci. Paris, Ser. I, 1997,  v. 325, 653-658;
P. Del Moral and A. Guionnet, Large deviations for interacting particle systems, Applications
to non linear filtering problems. Publications du Laboratoire de Statistiques et
Probabilités, Université Paul Sabatier, 1997, 05-97; P. Del Moral and A. Guionnet,
A central limit theorem for non linear filtering using interacting particle systems.
Publications du Laboratoire de Statistiques et Probabilités, Université Paul
Sabatier, 1997, 11-97; N.J. Gordon, D.J. Salmon and A.F.M. Smith, Novel approach to
non-linear/non-Gaussian Bayesian state estimation, IEE Proceedings, Part F, 1993, v. 140 (2),
107-113]) and were treated using different techniques. The algorithms involve the use of
a system of $n$ particles which evolve (mutate) in correlation with each other according to
law of a given Markov process and, at fixed times, give birth to a number of offsprings.
Several possible branching mechanisms are described and, after imposing some weak restrictions
on the branching mechanism, the empirical measure associated to the particle systems is proven
to converge (as $n$ tends to $\infty$) to a measure valued process which satisfies a two step
recurrence relation. Finally the result is applied to discrete time filtering. In this case
the limiting measure valued process is precisely conditional distribution of the signal given
the observation.
     Keywords: filtering, particle systems, branching algorithms, interacting algorithms,
measure valued processes, numerical solutions

L. Miclo
An Example of Application of Discrete Hardy's Inequalities
 pp. 319-330
      After having given a short proof of weighted Hardy's inequalities on N, adapted from
the continuous case, we show how to use them to evaluate, up to a universal factor, spectral
gaps and logarithmic Sobolev constants of birth and death processes on Z. We illustrate this
method by giving an example of such calculation.
      Keywords: birth and death Markov processes, spectral gaps, weighted Hardy's inequalities on discrete intervals

A. Alabert and M.A. Marmolejo
Reciprocal Property for a Class of Anticipating Stochastic Differential Equations
 pp. 331-356
      We study a class of one-dimensional stochastic differential equations with boundary
conditions by means of a change of variables that reduces the diffusion coefficient to a
constant. We obtain a representation of the type $X_t=G(t,Y_t)$, where $Y$ is the solution
of the simpler equation. This representation is used to show several properties of the
original equation. In particular, our main result is a characterization of the coefficients
for which the solution process satisfies a suitable Markov-type property, namely, the
reciprocal property.
      Keywords: anticipating stochastic differential equations, reciprocal processes



1999
Volume 5
Issue 4

C. Külske
(Non-) Gibbsianness and Phase Transitions in Random Lattice Spin Models
 pp. 357-383
     We consider disordered lattice spin models with finite-volume Gibbs measures
$\mu_{\L}[\eta](d\s)$. Here $\s$ denotes a lattice spin variable and $\eta$ a lattice random
variable with product distribution $\P$ describing the quenched disorder of the model. We ask:
when will the joint measures $\lim_{\L\uparrow\Z^d}\P(d\eta)\mu_{\L}[\eta](d\s)$ be [non-]
Gibbsian measures on the product of spin space and disorder space? We obtain general criteria
for both Gibbsianness and non-Gibbsianness providing an interesting link between phase
transitions at a fixed random configuration and Gibbsianness in product space: loosely
speaking, a discontinuity in the quenched Gibbs expectation can lead to non-Gibbsianness,
(only) if it can be observed on the spin observable  conjugate to the independent disorder
variables.
     Our main specific example is the random field Ising model in any dimension for which we
show almost sure [almost sure non-] Gibbsianness for the single- [multi-] phase region. We
also discuss models with disordered couplings, including spin glasses and random bond
ferromagnets, where various mechanisms are responsible for [non-] Gibbsianness.
     Keywords: disordered systems, Gibbs measures, non-Gibbsianness, random field model,
random bond model, spin glass

K. Ravishankar and L. Triolo
Diffusive Limit of the Lorentz Model with a Uniform Field Starting from the Markov Approximation
 pp. 385-421
     We study the motion of a charged particle moving under the influence of a uniform field
in the $x_1$ direction in ${\bf R}^d$. At exponential times with a parameter proportional to
the instantaneous speed the direction of motion of the particle is randomized while the  speed
is unchanged. This process describes the motion of a tracer particle in the Lorentz model with
uniform field in the Boltzmann - Grad limit. We prove the existence of the diffusive limit of
this random flight process with field for each value of initial energy, and characterize it
using the martingale problem method. We obtain the diffusion and drift coefficients as functions
of the initial energy.
     Keywords: diffusion, diffusive limit, Lorentz model, martingale problem

B. Gentz and M. Löwe
Fluctuations in the Hopfield Model at the Critical Temperature
 pp. 423-449
     We investigate the fluctuations of the order parameter in the Hopfield model of spin
glasses and neural networks at the critical temperature $1/\beta_\crit=1$. The number
of patterns $M(N)$ is allowed to grow with the number $N$ of spins but the growth rate is
subject either to the constraint $M(N)^{7}/N\to 0$ or even to the constraint
$M(N)^{13}/N\to 0$, depending on the precise formulation of the result. As the system size
$N$ increases, on a set of large probability the distribution of the appropriately scaled
order parameter under the Gibbs measure comes arbitrarily close (in a metric which generates
the weak topology) to a non-Gaussian measure which depends on the realization of the random
patterns. This random measure is given explicitly by its (random) density.
     Keywords: Hopfield model, spin glasses, neural networks, random disorder, limit theorems,
non-Gaussian fluctuations, critical temperature

S. Asmussen and J.F. Collamore
Exact Asymptotics for a Large Deviations Problem for the GI/G/1 Queue
 pp. 451-476
     Let $V$ be the steady-state workload and $Q$ the steady-state queue length in the GI/G/1
queue. We obtain the exact asymptotics for probabilities of the form
$\P\{V\ge a(t),\, Q\ge b(t)\}$ as $t\to\infty$. In the light-tailed case, there are three
regimes according  to the limiting value of $a(t)/b(t)$. Our analysis here extends and
simplifies recent work of Aspandiiarov and Pechersky [S. Aspandiiarov and E.A. Pechersky,
A large deviations problem for compound Poisson processes in queueing theory, Markov
Processes Relat. Fields, 1997, v.3, pp. 333-366]. In the heavy-tailed subexponential case,
a lower asymptotic bound is derived and shown to be the exact asymptotics in a regime where
$a(t)$, $b(t)$ vary in a certain way determined by the service time distribution.
     Keywords: change of measure, conditioned limit theorems, saddlepoint approximations,
subexponential distribution, workload

L. Mazliak
Approximation of a Partially Observable Stochastic Control Problem
 pp. 477-487
      We consider an approximation scheme for a discrete measure-valued control problem and
apply this technique to a stochastic control problem under partial observation.
      Keywords: filtering, stochastic control, partial observation

Contents of all issues 1995-1999
 pp. 487-495