Table of contents, Bernoulli Journal, vol. 8, no. 1 (2002)

Bootstrap of kernel smoothing in nonlinear time series; Authors: Jürgen Franke, Jens-Peter Kreiss, Enno Mammen

Prophet inequalities for optimal stopping rules with probabilistic recall; Authors: David Assaf, Ester Samuel-Cahn

Inverse method of images; Authors: Violet S.F. Lo, Gareth O. Roberts, Henry E. Daniels

Expansion of transition distributions of Lévy processes in small time; Authors: Ludger Rüschendorf, Jeannette H.C. Woerner

Identification and properties of real harmonizable fractional Lévy motions; Authors: Albert Benassi, Serge Cohen, Jacques Istas

A note on the equivalence of two approaches for specifying a Markov process; Authors: K.-S. Chan, H. Tong

On the existence or non-existence of solutions for certain backward stochastic differential equations; Authors: Jean-Pierre Lepeltier, Jaime San Martín

Editorial board, Bernoulli Journal, vol. 8, no. 1 (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

Kernel smoothing in nonparametric autoregressive schemes offers a powerful tool in modelling time series. We show that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resample or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.

Let X_i, i = 1, ..., n, be independent random variables, and consider an optimal stopping problem where an observation not chosen in the past is still available i steps later with some probability p_i, 1 ≥ p₁ ≥ ... ≥ p_{n -1} ≥ 0. Only one object may be chosen. After formulating the general solution to this optimal stopping problem, we consider `prophet inequalities' for this situation. Let V<sub>\bf p</sub> (X<sub>1</sub>, ..., X<sub>n</sub>) be the optimal value to the statistician. We show that for all non-trivial, non-negative X<sub>i</sub> and all n ≥ 2, the `ratio prophet inequality' \rm E[ \max (X₁, ..., X_n)] < (2 - p_{n -1} ) V_{\bf p} (X₁, ..., X_n) holds, and 2 - p_{n -1} is the `best constant'. This generalizes the classical prophet inequality with no recall, in which the best constant is 2. For any 0 ≤ X<sub>i</sub> ≤ 1, the `difference prophet inequality' \rm E[\max (X₁, ..., X_n)] - V_{\bf p} (X₁, ..., X_n) ≤ (1- p_n-1) [ 1 - (1 - p_{n - 1})^1/2 ]² / p²_n-1 holds. Prophet regions are also discussed.

We consider the problem of approximating the density of the time at which a Brownian path first crosses a curved boundary in cases where the exact density is not known or is difficult to compute. Approximation methods which involve the use of images will be proposed. These methods can be used not only for one-sided boundaries but also for the case of two-sided boundaries; not only for concave boundaries but also for convex boundaries. The square root boundary and parabolic boundary provide examples for numerical comparisons of the approximation methods.

We establish for small time t a series expansion of the transition density and the transition distribution function of Lévy processes in terms of the density and the spectral function of the Lévy measure, respectively. Furthermore, the integrals ith respect tothe distribution function weighted by 1/t are proved to converge to the integral ith respect tothe spectral function of the Lévy measure, when the integrated function does not increase too fast and t → 0.

The class of real harmonizable fractional Lévy motions (RHFLMs) is introduced. It is shown that these share many properties with fractional Brownian motion. These fields are locally asymptotically self-similar with a constant index H, and have Hölderian paths. Moreover, the identification of H for the RHFLMs can be performed with the so-called generalized variation method. Besides fractional Brownian motion, this class contains non-Gaussian fields that are asymptotically self-similar at infinity with a real harmonizable fractional stable motion of index \tilde{H} as tangent field. This last property should be useful in modelling phenomena with multiscale behaviour.

The probabilistic structure of a discrete-time (high-order) vector Markov process may be studied using two approaches. In the first approach, the Markov process is specified by the transition probability and the initial distribution. An alternative approach is via a stochastic difference equation. We have proved that these two approaches are equivalent under very mild conditions.

We investigate the existence of (local) solutions and explosions for backward stochastic differential equationswith generator | f(t,ω ,y,z)| ≤ G(y)+F(y)R(z), where G,F,R are continuous, G is increasing in {\mathbb R}₊ (decreasing in {\mathbb R}_-) and R is subquadratic. We study in detail the case f(t,ω ,y,z)=G(y)+A| z| ².