Table of contents, Bernoulli Journal, vol. 7, no. 4 (2001)

Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process; Authors: Alessandra Guglielmi, Richard L. Tweedie

Martingale convergence and the functional equation in the multi-type branching random walk; Authors: Andreas E. Kyprianou, A. Rahimzadeh Sani

Law of the iterated logarithm for random walks on nilpotent groups; Authors: Lucia Caramellino, Valentina Di Vincenzo

Addendum to `Asymptotic inference for a linear stochastic differential equation with time delay'; Authors: Alexander A. Gushchin, Uwe Küchler

Nonparametric regression with dependent errors; Authors: Yuhong Yang

Density estimation for spatial linear processes; Authors: Marc Hallin, Zudi Lu, Lanh Tat Tran

Testing additivity by kernel-based methods - what is a reasonable test?; Authors: Holger Dette, Carsten Von Lieres Und Wilkau

Editorial board, Bernoulli Journal, vol. 7, no. 4 (2001)

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

The distribution $\mathcal{M}_\alpha$ of the mean $\Gamma_\alpha$ of a Dirichlet process on the real line, with parameter $\alpha$, can be characterized as the invariant distribution of a real Markov chain $\Gamma_n$. In this paper we prove that, if $\alpha$ has finite expectation, the rate of convergence (in total variation) of $\Gamma_n$ to $\Gamma_\alpha$ is geometric. Upper bounds on the rate of convergence are found which seem effective, especially in the case where α has a support which is not doubly infinite. We use this to study an approximation procedure for $\mathcal{M}_\alpha$, and evaluate the approximation error in simulating $\mathcal{M}_\alpha$ using this chain. We include examples for a comparison with some of the existing procedures for approximating $\mathcal{M}_\alpha$, and show that the Markov chain approximation compares well with other methods.

A generalization of Biggins's martingale convergence theorem is proved for the multi-type branching random walk. The proof appeals to modern techniques involving the construction of size-biased measures on the space of marked trees generated by the branching process. As a simple consequence we obtain existence and uniqueness of solutions (within a specified class) to a system of functional equations.

We prove a law of the iterated logarithm for products of independent, identically distributed random variables taking values on simply connected, graded, nilpotent Lie groups as a consequence of similar results, of functional type, for stochastic processes which are solutions of ordinary differential equations with random coefficients.

We strengthen the convergence result proving the local asymptotic mixed normality property in one of the 11 cases considered in our previous paper.

We study minimax rates of convergence for nonparametric regression under a random design with dependent errors. It is shown that when the errors are independent of the explanatory variables, long-range dependence among the errors does not necessarily hurt regression estimation, which at first glance contradicts with earlier results by Hall and Hart, Wang, and Johnstone and Silverman under a fixed design. In fact we show that, in general, the minimax rate of convergence under the square $L_2$ loss is simply at the worse of two quantities: one determined by the massiveness of the class alone and the other by the severity of the dependence among the errors alone. The clear separation of the effects of the function class and dependence among the errors in determining the minimax rate of convergence is somewhat surprising. Examples of function classes under different covariance structures including both short- and long-range dependences are given.

The problem of estimating the marginal densities of a spatial linear process, observed over a grid of $\mathbb {Z}^N$, is considered. Under general conditions, kernel density estimators computed at any $k$-tuple of sites are shown to be asymptotically multivariate normal. Their limiting covariance matrix is also computed. Despite the huge development of nonparametric estimation methods in the analysis of time series data, little has so far been done to introduce them into the context of random fields. The generalization indeed is far from trivial since the points of $\mathbb {Z}^N$ do not have a natural ordering when $N>1$. No mixing conditions are required, but linearity is assumed.

In the common nonparametric regression model with high-dimensional predictor, several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are based either on the differences between a fit under the assumption of additivity and a fit in the general model, or on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L²-distance of the Nadaraya-Watson and the marginal integration estimator yields the (asymptotically) most efficient procedure, if these are compared with respect to the asymptotic behaviour under fixed and local alternatives. The finite-sample properties of the proposed procedures are investigated by means of a simulation study, which qualitatively confirms the asymptotic results.