Table of contents, Bernoulli Journal, vol. 7, no. 6 (2001)

Bias correction and bootstrap methods for a spatial sampling scheme; Authors: Peter Hall, Gavin Melville, Alan H. Welsh

Asymptotic behaviour of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1) errors; Authors: Milan Borkovec

Are classes of deterministic integrands for fractional Brownian motion on an interval complete?; Authors: Vladas Pipiras, Murad S. Taqqu

Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach; Authors: Emmanuel Gobet

Mixed fractional Brownian motion; Authors: Patrick Cheridito

Blocking measures for asymmetric exclusion processes via coupling; Authors: Pablo A. Ferrari, Joel L. Lebowitz, Eugene Speer

Editorial board, Bernoulli Journal, vol. 7, no. 6 (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

Motivated by sampling problems in forestry and related fields, we suggest a spatial sampling scheme for estimating the intensity of a point process. The technique is related to the `wandering quarter' method. In applications where the cost of identifying random points is high relative to the cost of taking measurements, for example when identification involves travelling within a large region, our approach has significant advantages over more traditional approaches such as T-square sampling. When the point process is Poisson we suggest a simple bias correction for a `naive' estimator of intensity, and also discuss a more complex estimator based on maximum likelihood. A technique for pivoting, founded on a fourth-root transformation, is proposed and shown to yield second-order accuracy when applied to construct bootstrap confidence intervals for intensity. Bootstrap methods for correcting edge effects and for addressing non-Poisson point-process models are also suggested.

We consider a stationary AR(1) process with ARCH(1) errors given by the stochastic difference equation

X_t=α X_{t-1}+\sqrt{β+λ X_{t-1}^2}\,ε_{t}\,, t∈{\mathbb{N}}\,,

where the (ε_t) are independent and identically distributed symmetric random variables. In contrast to ARCH and GARCH processes, AR(1) processes with ARCH(1) errors are not solutions of linear stochastic recurrence equations and there is no obvious way to embed them into such equations. However, we show that they still belong to the class of stationary sequences with regularly varying finite-dimensional distributions and therefore the theory of Davis and Mikosch can be applied. We present a complete analysis of the weak limit behaviour of the sample autocovariance and autocorrelation functions of (X_t), (|X_t|) and (X_t²). The results in this paper can be seen as a natural extension of results for ARCH(1) processes.