Table of contents, Bernoulli Journal, vol. 11, no. 3 (2005)

Extreme value theory for moving average processes with light-tailed innovations; Authors: Claudia Klüppelberg, Alexander Lindner

Exact asymptotics for estimating the marginal density of discretely observed diffusion processes; Authors: Cristina Butucea, Michael H. Neumann

Infinite divisibility and generalized subexponentiality; Authors: Takaaki Shimura, Toshiro Watanabe

Recurrent extensions of self-similar Markov processes and Cramér's condition; Authors: Víctor Rivero

Passage times for a spectrally negative Lévy process with applications to risk theory; Authors: Sung Nok Chiu, Chuancun Yin

On classifying processes; Authors: Gusztáv Morvai, Benjamin Weiss

On Hipp's compound Poisson approximations via concentration functions; Authors: Bero Roos

On the unlimited growth of a class of homogeneous multitype Markov chains; Authors: Miguel González, Rodrigo Martínez, Manuel Mota

Editorial board, Bernoulli Journal, vol. 11, no. 3 (2005)

Peter McCullagh, Editor-in-Chief

Peter J. Bickel, Editor

D.R. Cox, Editor

David Nualart, Editor

Edward Waymire, Editor

Lucien Birgé, Associate Editor

Evarist Giné, Associate Editor

Peter Hall, Associate Editor

Wilfrid S. Kendall, Associate Editor

Xiao-Li Meng, Associate Editor

Thomas Mikosch, Associate Editor

Gesine Reinert, Associate Editor

Yaacov Ritov, Associate Editor

David Siegmund, Associate Editor

Dag Tjøstheim, Associate Editor

Alexandre Tsybakov, Associate Editor

Sara van de Geer, Associate Editor

Eva Bjorn Vedel Jensen, Associate Editor

Alan Welsh, Associate Editor

Wei-Biao Wu, Associate Editor

Mitzi Nakatsuka, Editorial Assistant

Richard Leigh, Copy Editor

Ank Lepping, Desk Editor

Ole E. Barndorff-Nielsen, Past Editors-in-Chief(1995-2000)

Willem R. van Zwet, Past Editors-in-Chief(2000-2003)

Jens L. Jensen, Past Co-Editors (1995-2000)

Sara van de Geer, Past Co-Editors (2000-2003)

We consider stationary infinite moving average processes of the form $$Y_n=\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_i{Z}_{n+i},n\in \mathbb{Z},$$ where (Z_i)_i∈Z is a sequence of independent and identically distributed (i.i.d.) random variables with light tails and (c_i)_i∈Z is a sequence of positive and summable coefficients. By `light tails' we mean that Z₀ has a bounded density $f\left(t\right)\sim \nu \left(t\right)exp(-\psi (t\left)\right)$ , where ν(t) behaves roughly like a constant as t →∞ and ψ is strictly convex satisfying certain asymptotic regularity conditions. We show that the i.i.d. sequence associated with Y₀ is in the maximum domain of attraction of the Gumbel distribution. Under additional regular variation conditions on ψ, it is shown that the stationary sequence (Y_n)_n∈N has the same extremal behaviour as its associated i.i.d. sequence. This generalizes Rootzén's results where $f\left(t\right)\sim c{t}^{\alpha }exp(-t^p)\mathrm{for}c>0,\alpha \in R\mathrm{and}p>1$

We derive sharp asymptotic minimax bounds (that is, bounds which concern the exact asymptotic constant of the risk) for nonparametric density estimation based on discretely observed diffusion processes. We study two particular problems for which there already exist such results in the case of independent and identically distributed observations, namely, minimax density estimation in Sobolev classes with L₂-loss and in Hölder classes with L_∞-loss. We derive independently lower and upper bounds for the asymptotic minimax risks and show that they coincide with the classic efficiency bounds. We prove that these bounds can be attained by usual kernel density estimators. The lower bounds are obtained by analysing the problem of estimating the marginal density in certain families of processes, $\left\{\right\{X_i^f\},f\in {calF}_n\}$ , which are shrinking neighbourhoods of some central process, $\left\{X_i^{f_0}\right\}$ , in the sense that the set of densities {\cal F}_n forms a shrinking neighbourhood centred around f₀.

We introduce a new class of distributions by generalizing the subexponential class to investigate the asymptotic relation between the tails of an infinitely divisible distribution and its Lévy measure. We call a one-sided distribution μ O-subexponential if it has positive tail satisfying . Necessary and sufficient conditions for an infinitely divisible distribution to be O-subexponential are given in a similar way to the subexponential case in work by Embrechts et al. It is of critical importance that the O-subexponential is not closed under convolution roots. This property leads to the difference between our result and that corresponding to the subexponential class. Moreover, under the assumption that an infinitely divisible distribution has exponential tail, it is shown that an infinitely divisible distribution is convolution equivalent if and only if the ratio of its tail and its Lévy measure goes to a positive constant as x goes to infinity. Additionally, the upper and lower limits of the ratio of the tails of a semi-stable distribution and its Lévy measure are given.

Let ξ be a real-valued Lévy process that satisfies Cramér's condition, and X a self-similar Markov process associated with ξ via Lamperti's transformation. In this case, X has 0 as a trap and satisfies the assumptions set out by Vuolle-Apiala. We deduce from the latter that there exists a unique excursion measure \exc, compatible with the semigroup of X and such that \exc(X₀₊>0)=0. Here, we give a precise description of \exc via its associated entrance law. To this end, we construct a self-similar process X^\natural, which can be viewed as X conditioned never to hit 0, and then we construct \exc similarly to the way in which the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of \exc is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of \exc.

The distributions of the last passage time at a given level and the joint distributions of the last passage time, the first passage time and their difference for a general spectrally negative process are derived in the form of Laplace transforms. The results are applied to risk theory.

We prove several results concerning classifications, based on successive observations (X₁, ..., X_n) of an unknown stationary and ergodic process, for membership of a given class of processes, such as the class of all finite-order Markov chains.

This paper is devoted to a refinement of Hipp's method in the compound Poisson approximation to the distribution of the sum of independent but not necessarily identically distributed random variables. Approximations by related Kornya-Presman signed measures are also considered. By using alternative proofs, we show that several constants in the upper bounds for the Kolmogorov and the stop-loss distances can be reduced. Concentration functions play an important role in Hipp's method. Therefore, we provide an improvement of the constant in Le~Cam's bound for concentration functions of compound Poisson distributions. But we also follow Hipp's idea to estimate such concentration functions with the help of Kesten's concentration function bound for sums of independent random variables. In fact, under the assumption that the summands are identically distributed, we present a smaller constant in Kesten's bound, the proof of which is based on a slight sharpening of Le Cam's version of the Kolmogorov-Rogozin inequality.

We consider a homogeneous multitype Markov chain whose states have non-negative integer coordinates, and give criteria for deciding whether or not the chain grows indefinitely with positive probability. The results are applied to study the extinction problem in a general class of controlled multitype branching processes.