Table of contents, Bernoulli Journal, vol. 5, no. 6 (1999)

Limit laws for exponential families; Authors: August A. Balkema, Claudia Klüppelberg, Sidney I. Resnick

Revisiting conditional limit theorems for the mortal simple branching process; Authors: Anthony G. Pakes

Compound sums and subexponentiality; Authors: Hanspeter Schmidli

Symmetrization of binary random variables; Authors: Abram Kagan, Colin L. Mallows, Larry A. Shepp, Robert J. Vanderbei, Yehuda Vardi

Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution; Authors: Bero Roos

Some asymptotic properties of the local time of the uniform empirical process; Authors: Miklós Csörgö, Zhan Shi, Marc Yor

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

The asymptotic minimax constant for sup-norm loss in nonparametric density estimation; Authors: Alexander Korostelev, Michael Nussbaum

Bayesian smoothing in the estimation of the pair potential function of Gibbs point processes; Authors: Juha Heikkinen, Antti Penttinen

Editorial board, Bernoulli Journal, vol. 5, no. 6 (1999)

Ole E. Barndorff-Nielsen, Editor-in-Chief

Jens Ledet Jensen, Managing Co-Editor

Peter J. Bickel, Editor

David Nualart, Editor

David R. Cox, Editor

Edward C. Waymire, Editor

Søren Asmussen, Associate Editor

Adrian Baddeley, Associate Editor

Lucien Birgé, Associate Editor

Philippe Carmona, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Valentine Genon-Catalot, Associate Editor

Richard D. Gill, Associate Editor

Peter Guttorp, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Andrea Rotnitzky, Associate Editor

Terence P. Speed, Associate Editor

Dag Tjøstheim, Associate Editor

Richard L. Tweedie, Associate Editor

Andrew T.A. Wood, Associate Editor

Ank Lepping, Desk Editor

For a real random variable $X$ with distribution function $F$, define

$$\Lambda :=\{\lambda \in ℝ:K(\lambda ):=rmErm{e}^{\lambda X}<\mathrm{\infty }\}.$$

The distribution $F$ generates a natural exponential family of distribution functions $\{F_{\lambda },\lambda \in \Lambda \}$, where

$$rm{\mathrm{dF}}_{\lambda }\left(x\right):=rm{e}^{\lambda x}rm\mathrm{dF}\left(x\right)/K\left(\lambda \right),\lambda \in \Lambda .$$

We study the asymptotic behaviour of the distribution functions $F_{\lambda }$ as $\lambda$ increases to ${\lambda }_{\mathrm{\infty }}:=sup\Lambda$. If ${\lambda }_{\mathrm{\infty }}=\mathrm{\infty }$ then $F_{\lambda }\downarrow 0$ pointwise on $\left\{F<1\right\}$. It may still be possible to obtain a non-degenerate weak limit law $G\left(y\right)=lim{F}_{\lambda }(a_{\lambda }y+b_{\lambda })$ by choosing suitable scaling and centring constants $a_{\lambda }>0$ and $b_{\lambda }$, and in this case either $G$ is a Gaussian distribution or $G$ has a finite lower end-point $y_0=inf\left\{G>0\right\}$ and $G(y-y_0)$ is a gamma distribution. Similarly, if ${\lambda }_{\mathrm{\infty }}$ is finite and does not belong to $\Lambda$ then $G$ is a Gaussian distribution or $G$ has a finite upper end-point $y_{\mathrm{\infty }}$ and $1-G(y_{\mathrm{\infty }}-y)$ is a gamma distribution. The situation for sequences ${\lambda }_n\uparrow {\lambda }_{\mathrm{\infty }}$ is entirely different: any distribution function may occur as the weak limit of a sequence $F_{{\lambda }_n}(a_{\mathrm{nx}}+b_n)$.

The various conditional limit theorems for the simple branching process are considered within a unified setting. In the subcritical case conditioning events have the form $\{calH\in n+calS\}$, where $calH$ is the time to extinction, and $calS$ is a subset of the natural numbers. The resulting limit theorems contain all known forms, and collectively they are equivalent to the classical Yaglom form. In the critical case discrete limits exist provided $calS$ is a finite set. The principal results are extended to absorbing Markov chains. The Yaglom and Harris theorems for the critical case are generalized by considering the joint behaviour of generation sizes and total progeny conditioned by one-parameter families of events of the form $\{n<calH\le \alpha n\}$ and $\left\{calH>\alpha n\right\}$, where $1\le \alpha \le \mathrm{\infty }$. A simple representation of the marginal limit laws of the population sizes relates the Yaglom and Harris limits. Analogous structure is elucidated for the marginal limit laws of the total progeny.

We investigate compound distributions - for example, compound mixed Poisson distributions - in the case where the summands, the mixing distribution or the number of summands are subexponential. It is shown that such a distribution is subexponential. As an illustration the tail of the maximum of a Björk-Grandell process is obtained.

A random variable $Y$ is called an independent symmetrizer of a given random variable $X$ if (a) it is independent of $X$ and (b) the distribution of $X+Y$ is symmetric about $0$. In cases where the distribution of $X$ is symmetric about its mean, it is easy to see that the constant random variable $Y=\mathrm{EX}$ is a minimum-variance independent symmetrizer. Taking $Y$ to have the same distribution as $-X$ clearly produces a symmetric sum, but it may not be of minimum variance. We say that a random variable $X$ is symmetry resistant if the variance of any symmetrizer, $Y$, is never smaller than the variance of $X$. Let $X$ be a binary random variable: $P\{X=a\}=p$ and $P\{X=b\}=q$ where $a\ne b$, $0<p<1$, and $q=1-p$. We prove that such a binary random variable is symmetry resistant if (and only if) $p\ne 1/2$. Note that the minimum variance as a function of $p$ is discontinuous at $p=1/2$. Dropping the independence assumption, we show that the minimum variance reduces to $\mathrm{pq}-min(p,q)/2$, which is a continuous function of $p$.

The Poisson-binomial distribution is approximated by a Poisson law with respect to a new multi-metric (difference metric) unifying a broad class of probability metrics between discrete distributions. The accompanying non-metric situation is also considered leading to moderate- and large-deviation results. Using the Charlier B expansion and Fourier arguments, sharp bounds and asymptotic relations are given.

We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for the integral of the square of the local time. It is interesting to note that the corresponding problems for the Wiener process remain open. Properties of L^p-norms of the local time are studied. We also characterize the joint asymptotics of the local time at a fixed level and the maximum local time.

For the stochastic differential equation

$$dX\left(t\right)=aX\left(t\right)+bX(t-1),dt+dW\left(t\right),t\ge 0,$$

the local asymptotic properties of the likelihood function are studied. They depend strongly on the true value of the parameter $\theta =(a,b{)}^{*}$. Eleven different cases are possible if $\theta$ runs through ${ℝ}^2$. Let ${\hat{\theta }}_T$ be the maximum likelihood estimator of $\theta$ based on $\left(X\right(t),t\le T)$. Applications to the asymptotic behaviour of ${\hat{\theta }}_T$ as $T\to \mathrm{\infty }$ are given.

We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has already been found for nonparametric regression and for signal estimation in Gaussian white noise. Hölder classes for arbitrary smoothness index β>0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but in addition it depends on the maximum of densities in the function class.

A flexible Bayesian method is suggested for the pair potential estimation with a high-dimensional parameter space. The method is based on a Bayesian smoothing technique, commonly applied in statistical image analysis. For the calculation of the posterior mode estimator a new Monte Carlo algorithm is developed. The method is illustrated through examples with both real and simulated data, and its extension into truly nonparametric pair potential estimation is discussed.

Acknowledgement of refereeing work BJ vol. 05

Annual index BJ vol. 05