Table of contents, Bernoulli Journal, vol. 8, no. 3 (2002)

Coupling and Bernoullicity in random-cluster and Potts models; Authors: Olle Häggström, Johan Jonasson, Russell Lyons

A recursive method for functionals of Poisson processes; Authors: Dragan Banjevic, Hemant Ishwaran, Mahmoud Zarepour

Adaptive ranked-set sampling with multiple concomitant variables: an effective way to observational economy; Authors: Zehua Chen

Self-decomposability and Lévy processes in free probability; Authors: Ole E. Barndorff-Nielsen, Steen Thorbjørnsen

Recursive computation of the invariant distribution of a diffusion; Authors: Damien Lamberton, Gilles Pagès

On the moment problem; Authors: Allan Gut

Editorial board, Bernoulli Journal, vol. 8, no. 3 (2002)

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

An explicit coupling construction of random-cluster measures is presented. As one of the applications of the construction, the Potts model on amenable Cayley graphs is shown to exhibit at every temperature the mixing property known as Bernoullicity.

Functionals of Poisson processes arise in many statistical problems. They appear in problems involving heavy-tailed distributions in the study of limiting processes, while in Bayesian nonparametric statistics they are used as constructive representations for nonparametric priors. We describe a simple recursive method that is useful for characterizing Poisson process functionals and requires only the use of conditional probability. Applications of this technique to convex hulls, extremes, stable measures, infinitely divisible random variables and Bayesian nonparametric priors are discussed.

The notion of ranked-set sampling (RSS) proposed by McIntyre provides an effective means of achieving observational economy in certain particular situations. The use of concomitant variables broadens the range of application of RSS. In this paper, we deal with RSS with multiple concomitant variables and develop an adaptive RSS procedure in a general context. The particular case of multiple linear regression estimates with RSS is treated in detail. The procedure is illustrated with a real data set. Some simulation results are also given.

In this paper we study the bijection, introduced by Bercovici and Pata, between the classes of infinitely divisible probability measures in classical and in free probability. We prove certain algebraic and topological properties of that bijection (in the present paper denoted Λ), and those properties are then used to show, in particular, that Λ maps the class of classically self-decomposable probability measures onto the natural free counterpart which we define here. Further, we study Lévy processes in free probability and use the properties of Λ to construct stochastic integrals with respect to such processes. In particular, we derive the free analogue of the integral representation of self-decomposable random variables.

We propose a recursive stochastic algorithm with decreasing step to compute the invariant distribution $v$ of a Brownian diffusion process, in which we approximate $ν(f)$ for a wide class of possibly unbounded continuous functions $f$. We consider a somewhat general setting which includes cases where the diffusion may have several invariant distributions. Our main convergence result contains as a corollary the almost sure central limit theorem. Further, we investigate the weak rate of convergence of the algorithm. We show, in the class of polynomial steps$\gamma_n=n^{-\alpha}$, that it can be at most $n^{1/3}$ when the white noise has third moment zero and $n^{1/4}$ otherwise, where $n$ denotes the number of iterations of the algorithm.

The purpose of this paper is to provide some additional insight into the moment problem by connecting a condition by Lin, Bondesson's class of hyperbolically completely monotone densities, and the theory of regularly varying functions. In particular, two questions addressed in a recent paper by Stoyanov concerning powers of random variables and functions that (do not) preserve uniqueness will be investigated.