Table of contents, Bernoulli Journal, vol. 1, no. 1-2 (1995)

The foundation of the Bernoulli Society; Authors: David G. Kendall

Kernel estimation of relative risk; Authors: Julia E. Kelsall, Peter J. Diggle

Martingale estimation functions for discretely observed diffusion processes; Authors: Bo Martin Bibby, Michael Sørensen

On wavelet methods for estimating smooth functions; Authors: Peter Hall, Prakash Patil

Applications of the van Trees inequality: a Bayesian Cramér-Rao bound; Authors: Richard D. Gill, Boris Y. Levit

Time-scales for Gaussian approximation and its breakdown under a hierarchy of periodic spatial heterogeneities; Authors: Rabi N. Bhattacharya, Friedrich Götze

Explosive Poisson shot noise processes with applications to risk reserves; Authors: Claudia Klüppelberg, Thomas Mikosch

Quadratic covariation and an extension of Itô's formula; Authors: Hans Föllmer, Philip Protter, Albert N. Shiryayev

Limits of spatial branching populations(with Discussion); Authors: Anton Wakolbinger

Branching processes as population dynamics; Authors: Peter Jagers

Editorial board, Bernoulli Journal, vol. 1, no. 1-2 (1995)

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

Lucien Birgé, Associate Editor

Louis H. Y. Chen, Associate Editor

Hans Föllmer, Associate Editor

Peter McCullagh, Associate Editor

Alexander A. Novikov, Associate Editor

Yosihiko Ogata, Associate Editor

Victor Pérez-Abreu, Associate Editor

Dag Tjøstheim, Associate Editor

Introduction by the founding President of the Bernoulli Society

Estimation of a relative risk function using a ratio of two kernel density estimates is considered, concentrating on the problem of choosing the smoothing parameters. A cross-validation method is proposed, compared with a range of other methods and found to be an improvement when the actual risk is close to constant. In particular, theoretical and empirical comparisons demonstrate the advantage of choosing the smoothing parameters jointly. The methodology was motivated by a class of problems in environmental epidemiology, and an application in this area is described.

We consider three different martingale estimating functions based on discrete-time observations of a diffusion process. One is the discretized continuous-time score function adjusted by its compensator. The other two emerge naturally when optimality properties of the first are considered. Subject to natural regularity conditions, we show that all three martingale estimating functions result in consistent and asymptotically normally distributed estimators when the underlying diffusion is ergodic. Practical problems with implementing the estimation procedures are discussed through simulation studies of three specific examples. These studies also show that our estimators have good properties even for moderate sample sizes and that they are a considerable improvement compared with the estimator based on the unadjusted discretized continuous-time likelihood function, which can be seriously biased.

Without assuming any prior knowledge of wavelet methods, we develop theory describing their performance as estimators of smooth functions. The linear part of the wavelet estimator is discussed by analogy with classical kernel methods. Concise formulae are developed for its bias, variance and mean square error. These quantities oscillate somewhat erratically on a wavelength that is equivalent to the bandwidth, reflecting the irregular numerical fluctuations that are observed in practice. Nevertheless, the contributions of these oscillations to mean integrated square error tend to dampen one another out, even over very small intervals, with the result that mean integrated square error properties of linear wavelet methods are much closer to those of kernel methods than is perhaps reasonable, given the local behaviour. We illustrate the adaptive qualities of the nonlinear component of a wavelet estimator by describing its performance when the target function is smooth but has high-frequency oscillations. It is shown that the nonlinear component automatically adapts to changing local conditions, to the extent of achieving (except for a logarithmic factor) the same convergence rate as the optimal linear estimator, but without a need to adjust the underlying bandwidth. This makes explicitly clear the way in which the linear part of the estimator takes care of the `average' characteristics of the unknown curve, while the nonlinear part corrects for more erratic fluctuations, in a manner which is virtually independent of the construction of the linear part.

We use a Bayesian version of the Cramér-Rao lower bound due to van Trees to give an elementary proof that the limiting distribution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications.

The solution of the Itô equation $dX\left(t\right)=b\left\{X\right(t\left)\right\}dt+\beta \left\{X\right(t)/a\}dt+\sqrt{bD}dB\left(t\right)$ is analysed for $t\to \mathrm{\infty }$, $a\to \mathrm{\infty }$. In the range $1\ll t\ll a^{2/3}$, $X\left(t\right)$ is asymptotically Gaussian if $b$ is periodic, $\beta$ Lipschitzian; here the large-scale fluctuations may be ignored. In the range $t\gg a^2$, with both $b$ and $\beta$ periodic and divergence-free, $a$ integral, Gaussian approximation is again valid under an appropriate hypothesis on the geometry of $\beta$; here for some coordinates of $X\left(t\right)$ the dispersivity, or variance per unit time, may grow at the extreme rate $O\left(a^2\right)$ while stabilizing for others. As shown by examples, Gaussian approximation generally breaks down at intermediate time-scales. These results translate into asymptotics of a class of Fokker-Planck equations which arise in the prediction of contaminant transport in an aquifer under multiple scales of spatial heterogeneity. In particular, contrary to popular belief, the growth in dispersivity is always slower than linear.

We consider explosive Poisson shot noise processes as natural extensions of the classical compound Poisson process and investigate their asymptotic properties. Our main result is a functional central limit theorem with a self-similar Gaussian limit process which, in the classical case, is Brownian motion. The theorems are derived under regularity conditions on the moment and covariance functions of the shot noise process. The crucial condition is regular variation of the covariance function which implies the self-similarity of the limit process. The model is applied to delay in claim settlement in insurance portfolios. In this context we discuss some specific models and their properties. We also use the asymptotic theory for studying the ruin time and ruin probability for a risk process which is based on the Poisson shot noise process.

Let $X$ be a standard Brownian motion. We show that for any locally square integrable function $f$ the quadratic covariation $\left[f\right(X),X]$ exists as the usual limit of sums converging in probability. For an absolutely continuous function $F$ with derivative $f$, Itô's formula takes the form $F\left(X_t\right)=F\left(X_0\right)+{\int }_0^{t}f\left(X_s\right)d{X}_s+\frac{1}{2}\left[f\right(X),X{]}_t$. This is extended to the time-dependent case. As an example, we introduce the local time of Brownian motion at a continuous curve.

In critical branching and migrating populations, mobility of the individuals counteracts, and the clumping effect caused by the branching favours local extinction of the population in the large time limit. For example, d-dimensional critical binary branching Brownian motion (d>1) with a spatially inhomogeneous branching rate V(x), when started off with a homogeneous Poisson population, persists if V(x) ~ ||x||^d-2(log||x||)^-(1+ε), and suffers local extinction if V(x) ~ ||x||^d-2(log||x||)^-1 as ||x||→∞; this can be derived from a probabilistic persistence criterion (Theorem 2). Besides presenting this result, the paper reviews conditions on the parameters of various other models which are necessary and sufficient for persistence, and discusses related results for superprocesses. Common to the proofs is the method of analysing the backward tree of an individual encountered in an old population, originally due to Kallenberg (1977) and Liemant (1981) in discrete time settings.

Branching processes were once born out of a question from (human) population dynamics. Lately the driving forces have been, and continue to be, more of pure mathematical nature. Nevertheless, the resulting theory turns out to solve many classical problems from general, usually deterministic, population dynamics. These will be reviewed, with an emphasis on basic structure and on problems of the rate of population growth and the ensuing population composition. Special attention will be paid to possible interaction between individuals, or between the environment or population as a whole and individual reproduction behaviour. But the framework will remain the general model without explicit special assumptions about the form of interactions, lifespan distribution or reproduction.