Table of contents, Bernoulli Journal, vol. 9, no. 1 (2003)

Hitting, occupation and inverse local times of one-dimensional diffusions: martingale and excursion approaches authors: Jim Pitman , Marc Yor

Explicit formulae for time-space Brownian chaos authors: Giovanni Peccati

A note on the Laplace--Varadhan integral lemma authors: P. Del Moral , T. Zajic

Moderate deviations for mean-field Gibbs measures authors: Peter Eichelsbacher , Tim Zajic

Wavelet analysis of conservative cascades authors: Sidney Resnick , Gennady Samorodnitsky , Anna Gilbert , Walter Willinger

Semi-parametric efficiency, distribution-freeness and invariance authors: Marc Hallin , Bas J.M. Werker

Parametrization-invariant Wald tests authors: P.V. Larsen , P.E. Jupp

Editorial board, Bernoulli Journal, vol. 9, no. 1 (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

Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process $X$, first discussed by It\^o and McKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on $L_T^y$, the local time of $X$ at level $y$ before a suitable random time $T$, yield formulae for the joint Laplace transform of $L_T^y$ and the times spent by $X$ above and below level $y$ up to time $T$.

Let $F$ be a square-integrable and infinitely weakly differentiable functional of a standard Brownian motion $X$: we show that the $n$th integrand in the time-space chaotic decomposition of $F$ has the form $\mathbb{E}\lambdaeft(\alphapha _{\lambdaeft( n\right)}D^{n}F\mid X_{t_{1}},\rm dots,X_{t_{n}}\right)$, where $\alphapha _{\lambdaeft( n\right)}$ is a transform of Hardy type and $D^{n}$ denotes the $n$th derivative operator. In this way, we complete the results of previous papers, and provide a time-space counterpart to the classic Stroock formulae for Wiener chaos. Our main tool is an extension of the Clark--Ocone formula in the context of initially enlarged filtrations. We discuss an application to the static hedging of path-dependent options in a continuous-time financial model driven by $X$. A formal connection between our results and the orthogonal decomposition of the space of square-integrable functionals of a standard Brownian bridge -- as proved by Gosselin and Wurzbacher -- is also established.

We propose a complement to the Laplace--Varadhan integral lemma arising in the large-deviations literature. We examine a situation in which the state space may depend on the rate of deviations. We use this framework to discuss large-deviations principles for an abstract class of discrete generation interacting particle systems as well as for pure jump and McKean--Vlasov diffusions.

We present a moderate-deviations principle around non-degenerate attractors of the empirical measure of random variables distributed according to a mean-field Gibbs measure. We state a result for a large class of densities of the Gibbs measure. This result is an application of a rank-dependent moderate-deviations principle for a collection of $U$-empirical measures. The results are applied for diffusion processes with mean-field interaction leading to a McKean--Vlasov limit, and to the Curie--Weiss model.

A conservative cascade is an iterative process that fragments a given set into smaller and smaller pieces according to a rule which preserves the total mass of the initial set at each stage of the construction almost surely and not just in expectation. Motivated by the importance of conservative cascades in analysing multifractal behaviour of measured Internet traffic traces, we consider wavelet-based statistical techniques for inference about the cascade generator, the random mechanism determining the redistribution of the set's mass at each iteration. We provide two estimators of the structure function, one asymptotically biased and one not, and prove consistency and asymptotic normality in a range of values of the argument of the structure function less than a critical value. Simulation experiments illustrate the asymptotic properties of these estimators for values of the argument both below and above the critical value. Beyond the critical value, the estimators are shown not to be asymptotically consistent.

Semi-parametric models typically involve a finite-dimensional parameter of interest ${\pmb\theta}\in{\pmb\Theta}\subseteq\mathbb{R}^k$, along with an infinite-dimensional nuisance parameter~$f$. Quite often, the submodels corresponding to a fixed value of ${\pmb\theta}$ possess a group structure that induces a maximal invariant $\sigma$-field ${\cal B}({\pmb\theta})$. In classical examples, where $f$ denotes the density of some independ\-ent and identically distributed innovations, ${\cal B}({\pmb\theta})$ is the $\sigma$-field generated by the ranks of the residuals associated with the parameter value ${\pmb\theta}$. It is shown that semi-parametrically efficient distribution-free inference procedures can generally be constructed from parametrically optimal ones by conditioning on ${\cal B}({\pmb\theta})$; this implies, for instance, that semi-parametric efficiency (at given $\pmb\theta$ and $f$) can be attained by means of rank-based methods. The same procedures, when combined with a consistent estimation of the underlying nuis\-ance density $f$, yield conditionally distribution-free semi-parametrically efficient inference methods, for example, semi-parametrically efficient permutation tests. Remarkably, this is achieved without any explicit tangent space or efficient score computations, and without any sample-splitting device. By means of several examples, including both i.i.d. and time-series models, we show how these results apply in models for which rank-based inference or permutation tests have so far seldom been considered.

Parametrization-invariant versions of Wald statistics are introduced, based on yoke geo\-metry. The construction is illustrated by several examples. In the case of simple null hypotheses, formulae are given for generalized Bartlett corrections of these geometric Wald statistics which bring their null distributions close to their large-sample asymptotic distributions.