Table of contents, Bernoulli Journal, vol. 7, no. 5 (2001)

Adaptive estimation of the fractional differencing coefficient; Authors: Anatoli Iouditsky, Eric Moulines, Philippe Soulier

Local polynomial estimation with a FARIMA-GARCH error process; Authors: Jan Beran, Yuanhua Feng

Confidence intervals for the tail index; Authors: Shihong Cheng, Liang Peng

Backward stochastic differential equations and Feynman-Kac formula for Lévy processes, with applications in finance; Authors: David Nualart, Wim Schoutens

Cahn-Hilliard stochastic equation: existence of the solution and of its density; Authors: Caroline Cardon-Weber

The Riemann zeta distribution; Authors: Gwo Dong Lin, Chin-Yuan Hu

Editorial board, Bernoulli Journal, vol. 7, no. 5 (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

Semi-parametric estimation of the fractional differencing coefficient $d$ of a long-range dependent stationary time series has received substantial attention in recent years. Some of the so-called local estimators introduced early on were proved rate-optimal over relevant classes of spectral densities. The rates of convergence of these estimators are limited to $n^{2/5}$, where $n$ is the sample size. This paper focuses on the fractional exponential (FEXP) or broadband estimator of $d$. Minimax rates of convergence over classes of spectral densities which are smooth outside the zero frequency are obtained, and the FEXP estimator is proved rate-optimal over these classes. On a certain functional class which contains the spectral densities of FARIMA processes, the rate of convergence of the FEXP estimator is $(n$/log$(n))^{1/2}$, thus making it a reasonable alternative to parametric estimators. As usual in semi-parametric estimation problems, these rate-optimal estimators are infeasible, since they depend on an unknown smoothness parameter defining the functional class. A feasible adaptive version of the broadband estimator is constructed. It is shown that this estimator is minimax rate-optimal up to a factor proportional to the logarithm of the sample size.

This paper considers estimation of the trend function $g$ as well as its $\nu$th derivative $g^{(\nu)}$ in a so-called semi-parametric FARIMA-GARCH model by local polynomial fits. The focus is on the derivation of the asymptotic normality of $\hat g^{(\nu)}$. A central limit theorem based on martingale theory is developed. Asymptotic normality of the sample mean of a FARIMA-GARCH process is proved. These results are then used to show the asymptotic normality of $\hat g^{(\nu)}$. As an auxiliary result, the weak consistency of a weighted sum is obtained for second-order stationary time series with short or long memory under very weak conditions. Formulae for the mean integrated square error and the asymptotically optimal bandwidth of $\hat g^{(\nu)}$ are also given..

One of the best-known estimators for the tail index of a heavy-tailed distribution is the Hill estimator. In this paper, confidence intervals based on the asymptotic normal approximation of the Hill estimator are studied. The coverage accuracy is evaluated and the theoretical optimal choice of the sample fraction for the one-sided confidence interval is given. One surprising finding is that the order of optimal coverage accuracy for the one-sided confidence interval depends on the sign of the second-order regular variation.

In this paper we show the existence and uniqueness of a solution for backward stochastic differential equations driven by a Lévy process with moments of all orders. The results are important from a pure mathematical point of view as well as in finance: an application to Clark-Ocone and Feynman-Kac formulas for Lévy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equation provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a Lévy market.

We show the existence and uniqueness of a function-valued process solution to the stochastic Cahn-Hilliard equation driven by space-time white noise with a nonlinear diffusion coefficient. Then we show that the solution is locally differentiable in the sense of the Malliavin calculus, and, under some non-degeneracy condition on the diffusion coefficient, that the law of the solution is absolutely continuous with respect to Lebesgue measure.

Let $\zeta$ be the Riemann zeta function. Khinchine (1938) proved that the function $f_\sigma(t)=\zeta(\sigma + $i $t)/\zeta(\sigma)$, where $\sigma > 1$ and $t$ is real, is an infinitely divisible characteristic function. We investigate further the fundamental properties of the corresponding distribution of $f_\sigma$, the Riemann zeta distribution, including its support and unimodality. In particular, the Riemann zeta random variable is represented as a linear function of infinitely many independent geometric random variables. To extend Khinchine's result, we construct the Dirichlet-type characteristic functions of discrete distributions and provide a sufficient condition for the infinite divisibility of these characteristic functions. By way of applications, we give probabilistic proofs for some identities in number theory, including a new identity for the reciprocal of the Riemann zeta function.