Table of contents, Bernoulli Journal, vol. 5, no. 1 (1999)

Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists; Authors: David J. Aldous

Uniform oscillations of the local time of iterated Brownian motion; Authors: Nathalie Eisenbaum, Zhan Shi

On the growth of variances in a central limit theorem for strongly mixing sequences; Authors: Richard C. Bradley

Large deviations and variational theorems for marginal problems; Authors: Patrick Cattiaux, Fabrice Gamboa

Reversible Markov chains and optimality of symmetrized empirical estimators; Authors: Priscilla E. Greenwood, Wolfgang Wefelmeyer

Prequential probability: principles and properties; Authors: A. Philip Dawid, Vladimir G. Vovk

On the relationship between α connections and the asymptotic properties of predictive distributions; Authors: José M. Corcuera, Federica Giummolè

Estimating the end-point of a probability distribution using minimum-distance methods; Authors: Peter Hall, Julian Z. Wang

Editorial board, Bernoulli Journal, vol. 5, no. 1 (1999)

Ole E. Barndorff-Nielsen, Editor-in-Chief

Jens Ledet Jensen, Managing Co-Editor

Peter J. Bickel, Editor

David Nualart, Editor

David R. Cox, Editor

Edward C. Waymire, Editor

Søren Asmussen, Associate Editor

Adrian Baddeley, Associate Editor

Lucien Birgé, Associate Editor

Philippe Carmona, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Valentine Genon-Catalot, Associate Editor

Richard D. Gill, Associate Editor

Peter Guttorp, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Andrea Rotnitzky, Associate Editor

Terence P. Speed, Associate Editor

Dag Tjøstheim, Associate Editor

Richard L. Tweedie, Associate Editor

Andrew T.A. Wood, Associate Editor

Ank Lepping, Desk Editor

Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at stochastic rate K(x,y)/N, where K is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x,y)=1 and K(x,y)=xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.

Burdzy and Khoshnevisan in 1995, Csáki et al. in 1996 and Xiao in 1997 have given some interesting information about the modulus of continuity of the local time of iterated Brownian motion. The aim of this paper is to provide the exact rate functions, respectively for the modulus of continuity and for the modulus of non-differentiability. Our approach strongly relies on Ray-Knight theorems for Brownian local times and on fine properties of Bessel processes.

In a central limit theorem under certain strong mixing conditions, one does not quite have an asymptotic linear growth of the variance of the partial sums.

On a product probability space (E ×F, P), we give variational characterizations for the existence of a probability measure Q with given marginals, such that Q is absolutely continuous with respect to P and its density satisfies some integrability conditions. These characterizations, which are in some sense the dual formulation of a theorem due to Strassen, are obtained by using large-deviations methods. We also study the minimal realizations of such Q.

Suppose that we want to estimate the expectation of a function of two arguments under the stationary distribution of two successive observations of a reversible Markov chain. Then the usual empirical estimator can be improved by symmetrizing. We show that the symmetrized estimator is efficient. We point out applications to discretely observed continuous-time processes. The proof is based on a result for general Markov chain models which can be used to characterize efficient estimators in any model defined by restrictions on the stationary distribution of a single or two successive observations.

Forecaster has to predict, sequentially, a string of uncertain quantities $(X_1,X_2,...)$, whose values are determined and revealed, one by one, by Nature. Various criteria may be proposed to assess Forecaster's empirical performance. The weak prequential principle requires that such a criterion should depend on Forecaster's behaviour or strategy only through the actual forecasts issued. A wide variety of appealing criteria are shown to respect this principle. We further show that many such criteria also obey the strong prequential principle, which requires that, when both Nature and Forecaster make their choices in accordance with a common joint distribution $P$ for $(X_1,X_2,...)$, certain stochastic properties, underlying and justifying the criterion and inferences based on it, hold regardless of the detailed specification of $P$. In order to understand further this compliant behaviour, we introduce the prequential framework, a game-theoretic basis for probability theory in which it is impossible to violate the prequential principles, and we describe its connections with classical probability theory. In this framework, in order to show that some criterion for assessing Forecaster's empirical performance is valid, we have to exhibit a winning strategy for a third player, Statistician, in a certain perfect-information game. We demonstrate that many performance criteria can be formulated and are valid in the framework and, therefore, satisfy both prequential principles.

In a recent paper, Komaki studied the second-order asymptotic properties of predictive distributions, using the Kullback-Leibler divergence as a loss function. He showed that estimative distributions with asymptotically efficient estimators can be improved by predictive distributions that do not belong to the model. The model is assumed to be a multidimensional curved exponential family. In this paper we generalize the result assuming as a loss function any f divergence. A relationship arises between α connections and optimal predictive distributions. In particular, using an α divergence to measure the goodness of a predictive distribution, the optimal shift of the estimate distribution is related to α-covariant derivatives. The expression that we obtain for the asymptotic risk is also useful to study the higher-order asymptotic properties of an estimator, in the mentioned class of loss functions.

A technique based on minimum distance, derived from a coefficient of determination and representable in terms of Greenwood's statistic, is used to derive an estimator of the end-point of a distribution. It is appropriate in cases where the actual sample size is very large and perhaps unknown. The minimum-distance estimator is compared with a competitor based on maximum likelihood and shown to enjoy lower asymptotic variance for a range of values of the extremal exponent. When only a small number of extremes is available, it is well defined much more frequently than the maximum-likelihood estimator. The minimum-distance method allows exact interval estimation, since the version of Greenwood's statistic on which it is based does not depend on nuisance parameters.