Table of contents, Bernoulli Journal, vol. 8, no. 6 (2002)

Concentration and deviation inequalities in infinite dimensions via covariance representations ; Authors: Christian Houdré , Nicolas Privault

An extension of Sanov's theorem: application to the Gibbs conditioning principle ; Authors: Christian Léonard , Jamal Najim

The first exit time of Brownian motion from a parabolic domain ; Authors: Mikhail Lifshits , Zhan Shi

A technique for exponential change of measure for Markov processes ; Authors: Zbigniew Palmowski , Tomasz Rolski

Optimal procedures based on interdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against ARMA dependence ; Authors: Marc Hallin , Davy Paindaveine

Marginalizing and conditioning in graphical models ; Authors: Jan T.A. Koster

Editorial board, Bernoulli Journal, vol. 8, no. 6 (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

Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative to the use of logarithmic Sobolev inequalities. It enables the recovery of known concentration and deviation inequalities on the Wiener and Poisson space (including those given by sharp logarithmic Sobolev inequalities), and extends results available in the discrete case, i.e. on the infinite cube $\{-1,1\}^\infty$.

A large-deviation principle is proved for the empirical measures of independent and ident\-ically distributed random variables with a topology based on functions having only some exponential moments. The rate function differs from the usual relative entropy: it involves linear forms which are no longer measures. Following Stroock and Zeitouni, the Gibbs conditioning principle (GCP) is then derived with the help of the previous result. Apart from a more direct proof than has previously been available, the main improvements with respect to GCPs already published are the following: convergence holds in situations where the underlying log-Laplace transform (the pressure) may not be steep and the constraints are built on energy functions admitting only some finite exponential moments. Basic techniques from the theory of Orlicz spaces appear to be a powerful tool.

Consider a planar Brownian motion starting at an interior point of the parabolic domain $D=\{ (x,y): \; y>x^2\}$, and let $\tau_D$ denote the first time the Brownian motion exits from $D$. The tail behaviour (or equivalently, the integrability property) of $\tau_D$ is somewhat exotic since it arises from an interference of large-deviation and small-deviation events. Our main result implies that the limit of $T^{-1/3} \log\mathbb{P}\{ \tau_D >T\}$, $T\to \infty$, exists and equals $-3\pi^2/8$, thus improving previous estimates by Bañuelos et al. and Li. The existence of the limit is proved by applying the classical Schilder large-deviation theorem. The identification of the limit leads to a variational problem, which is solved by exploiting a theorem of Biane and Yor relating different additive functionals of Bessel processes. Our result actually applies to more general parabolic domains in any (finite) dimension.

We consider a Markov process $X(t)$ with extended generator ${\mathbf{A}}$ and domain $\cal{D}({\mathbf{A}})$. Let $\{{\cal F}_t\}$ be a right-continuous history filtration and ${\mathbb{P}}_t$ denote the restriction of ${\mathbb{P}}$ to ${\cal F}_t$. Let $\tilde{{\mathbb{P}}}$ be another probability measure on $(\Omega,{\cal F})$ such that $\rm d\tilde{{\mathbb{P}}}_t/\rm d{\mathbb{P}}_t=E^h(t)$, where $$ E^h(t)=\frac{h(X(t))}{h(X(0))}\exp \left(-\int_0^t \frac{({\mathbf{A}}h)(X(s))}{h(X(s))}\rm ds \right) $$ is a true martingale for a positive function $h\in\cal{D}({\mathbf{A}})$. We demonstrate that the process $X(t)$ is a Markov process on the probability space $ (\Omega, {\cal F}, \{{\cal F}_t\},\tilde{{\mathbb{P}}}) $, we find its extended generator $\tilde{{\mathbf{A}}}$ and provide sufficient conditions under which $\cal{D}(\tilde{{\mathbf{A}}})=\cal{D}({\mathbf{A}})$. We apply this result to continuous-time Markov chains, to piecewise deterministic Markov processes and to diffusion processes (in this case a special choice of $h$ yields the classical Cameron--Martin--Girsanov theorem).

We propose a multivariate generalization of signed-rank tests for testing elliptically symmetric white noise against ARMA serial dependence. These tests are based on Randles's concept of interdirections and the ranks of pseudo-Mahalanobis distances. They are affine-invariant and asymptotically equivalent to strictly distribution-free statistics. Depending on the score function considered (van der Waerden, Laplace. $\ldots$), they allow for locally asymptotically maximin tests at selected densities (multivariate normal, multivariate double exponential, $\ldots$). Local powers and asymptotic relative efficiencies with respect to the Gaussian procedure are derived. We extend to the multivariate serial context the Chernoff--Savage result, showing that classical correlogram-based procedures are uniformly dominated by the van der Waerden version of our tests, so that correlogram methods are not admissible in the Pitman sense. We also prove an extension of the celebrated Hodges--Lehmann `.864 result', providing, for any fixed space dimension, the lower bound for the asymptotic relative efficiency of the proposed multivariate Spearman type tests with respect to the Gaussian tests. These asymptotic results are confirmed by a Monte Carlo investigation.

A class of graphs is introduced which is closed under marginalizing and conditioning. It is shown that these operations can be executed by performing in arbitrary order a sequence of simple, strictly local operations on the graph at hand. The results are based on a simplification of J. Pearl's notion of $d$-separation. As the simplification does not change the separation properties of graphs for which the original $d$-separation concept is applicable (e.g., directed graphs), it constitutes a true generalization of the latter concept to the present class of graphs.