Table of contents, Bernoulli Journal, vol. 9, no. 6 (2003)

Large deviations in the Langevin dynamics of a short-range spin glass ; Authors: Gérard Ben Arous , Michel Sortais

An S-transform approach to integration with respect to a fractional Brownian motion ; Authors: Christian Bender

Baum--Katz laws for certain weighted sums of independent and identically distributed random variables ; Authors: Hartmut Lanzinger , Ulrich Stadtmüller

A quantization algorithm for solving multidimensional discrete-time optimal stopping problems ; Authors: Vlad Bally , Gilles Pagès

Small-diffusion asymptotics for discretely sampled stochastic differential equations ; Authors: Michael Sørensen , Masayuki Uchida

Parameter estimation for the supercritical contact process ; Authors: Marta Fiocco , Willem R. Van Zwet

Markov bases for decomposable graphical models ; Authors: Adrian Dobra

Editorial board, Bernoulli Journal, vol. 9, no. 6 (2003)

Willem R. van Zwet,Editor-in-Chief

Sara A. van de Geer,Co-Editor

Peter J. Bickel,Editor

D.R. Cox,Editor

David Nualart,Editor

Edward C. Waymire,Editor

Adrian Baddeley,Associate Editor

Lucien Birgé,Associate Editor

Phillipe Carmona,Associate Editor

Paul Embrechts,Associate Editor

Evarist Giné,Associate Editor

Friedrich Götze,Associate Editor

Peter Guttorp,Associate Editor

Frank den Hollander,Associate Editor

Iain Johnstone,Associate Editor

Wilfrid S. Kendall,Associate Editor

M.R. Leadbetter,Associate Editor

Enno Mammen,Associate Editor

Peter McCullagh,Associate Editor

Thomas Mikosch,Associate Editor

Dominique Picard,Associate Editor

David Siegmund,Associate Editor

Simon Tavaré,Associate Editor

Dag Tjøstheim,Associate Editor

Tineke H. Bakker,Editorial Assistant

Richard Leigh,Copy Editor

Ank Lepping,Desk Editor

Ole E. Barndorff-Nielsen (1995-2000),Past Editor-in-Chief

Jens L. Jensen (1995-2000),Past Managing Co-editor

We consider a Langevin dynamics associated with a $d$-dimensional Edwards--Anderson model having Gaussian coupling variables, and show that the averaged law of the empirical process satisfies a large-deviation principle according to a good rate functional $\mathcal{I}^a$ having a unique minimizer $Q_{\infty}$. The asymptotic dynamics $Q_{\infty}$ may be characterized as the unique weak solution corresponding to a non-Markovian system of interacting diffusions having an infinite range of interaction. We then establish that the quenched law of the empirical process also obeys a large-deviation principle, according to a (deterministic) good rate functional $\mathcal{I}^q$ satisfying $\mathcal{I}^q\geq \mathcal{I}^a$, so that, for a typical realization of the disorder variables, the quenched law of the empirical process also converges exponentially fast to a Dirac mass concentrated at $Q_{\infty}$.

We give an elementary definition of the (Wick--)Itô integral with respect to a fractional Brownian motion using the expectation, the ordinary Lebesgue integral and the classical (simple) Wiener integral. Then we provide new and simple proofs of some basic properties of this integral, including the so-called fractional Itô isometry. We calculate the expectation of the fractional Itô integral under change of measure and prove a Girsanov theorem for the fractional Itô integral (not only for fractional Brownian motion). We then derive an Itô formula for functionals of a fractional Wiener integral. Finally, we compare our approach with other approaches that yield essentially the same integral.

We consider weighted sums $\sum_k p_{nk}\, X_k$ of independent and identically distributed random variables $(X_n)$ and compare the tail probabilities of these sums with the moment conditions on $X_1$, that is, we prove various results of Baum--Katz type. Some special examples of weights $p_{nk}$ originating from summability are discussed.

A new grid method for computing the Snell envelope of a function of an $\mathbb{R}^d$-valued simulatable Markov chain $(X_k)_{0\lambda \leq k\lambda \leq n}$ is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every $X_k$ is replaced by a `quantized approximation' $\widehat{X}_k$ taking its values in a grid $\Gamma_k$ of size $N_k$. The $n$ grids and their trans\-ition probability matrices form a discrete tree on which a pseudo-Snell envelope is devised by mimicking the regular dynamic programming formula. Using the quantization theory of random vectors, we show the existence of a set of optimal grids, given the total number $N$ of elementary $\mathbb{R}^d$-valued quantizers. A recursive stochastic gradient algorithm, based on simulations of $(X_k)_{0\lambda \leq k \lambda \leq n}$, yields these optimal grids and their transition probability matrices. Some a priori error estimates based on the $L^p$-quantization errors $\|X_k-\widehat X_k\|_{_p}$ are established. These results are applied to the computation of the Snell envelope of a diffusion approximated by its (Gaussian) Euler scheme. We apply these result to provide a discretization scheme for reflected backward stochastic differential equations. Finally, a numerical experiment is carried out on a two-dimensional American option pricing problem.

1051-1069: The minimum-contrast estimation of drift and diffusion coefficient parameters for a multi-dimensional diffusion process with a small dispersion parameter $\varepsilon$ based on a Gaussian approximation to transition density is presented when the sample path is observed at equidistant times $k/n$, $k=0,1,\rm \dots,n$. We study asymptotic results for the minimum-contrast estimator as $\varepsilon$ goes to $0$ and $n$ goes to $\infty$ simultaneously.

Contact processes -- and, more generally, interacting particle processes -- can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. We begin a statistical study of the supercritical contact process that starts with a single infected site at the origin and is conditioned on survival of the infection. We consider the statistical problem of estimating the parameter $\lambda$ of the process on the basis of an observation of the process at a single time $t$. We propose an estimator of $\lambda$ and show that it is consistent and asymptotically normal as $t \rightarrow \infty$.

We show that primitive data swaps or moves are the only moves that have to be included in a Markov basis that links all the contingency tables having a set of fixed marginals when this set of marginals induces a decomposable independence graph. We give formulae that fully identify such Markov bases and show how to use these formulae to dynamically generate random moves.