Table of contents, Bernoulli Journal, vol. 12, no. 1 (2006)

Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations; Authors: Ole E. Barndorff-Nielsen, Makoto Maejima, Ken-Iti Sato

On sampling distributions for coalescent processes with simultaneous multiple collisions; Authors: M. Möhle

The shape of a sequence of dual random triangles; Authors: John Gates

Large deviations of the kernel density estimator in L¹(R^d) for reversible Markov processes; Authors: Liangzhen Lei

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H>½; Authors: Marco Ferrante, Carles Rovira

The reversible nearest particle system on a finite interval; Authors: Dayue Chen, Juxin Liu, Fuxi Zhang

Brownian motion in self-similar domains; Authors: Dante Deblassie, Robert Smits

On Kesten's counterexample to the Cramér-Wold device for regular variation; Authors: Henrik Hult, Filip Lindskog

A continuous Gaussian approximation to a nonparametric regression in two dimensions; Authors: Andrew V. Carter

Multivariate prediction; Authors: José Manuel Corcuera, Federica Giummolè

Which multivariate gamma distributions are infinitely divisible?; Authors: Philippe Bernardoff

Central limit theorem and convergence to stable laws in Mallows distance; Authors: Oliver Johnson, Richard Samworth

Editorial board, Bernoulli Journal, vol. 12, no. 1 (2006)

Peter McCullagh, Editor-in-Chief

Peter J. Bickel, Editor

D.R. Cox, Editor

David Nualart, Editor

Edward Waymire, Editor

David Applebaum, Associate Editor

Lucien Birgé, Associate Editor

Evarist Giné, Associate Editor

Peter Hall, Associate Editor

Wilfrid S. Kendall, Associate Editor

Thomas Mikosch, Associate Editor

Jesper Møller, Associate Editor

Gesine Reinert, Associate Editor

Ya'acov Ritov, Associate Editor

David Siegmund, Associate Editor

Dag Tjøstheim, Associate Editor

Alexandre Tsybakov, Associate Editor

Sara van de Geer, Associate Editor

Eva Bjorn Vedel Jensen, Associate Editor

Alan Welsh, Associate Editor

Wei-Biao Wu, Associate Editor

Mitzi Nakatsuka, Editorial Assistant

Richard Leigh, Copy Editor

Ank Lepping, Desk Editor

Ole E. Barndorff-Nielsen, Past Editors-in-Chief(1995-2000)

Willem R. van Zwet, Past Editors-in-Chief(2000-2003)

Jens L. Jensen, Past Co-Editors (1995-2000)

Sara van de Geer, Past Co-Editors (2000-2003)

The class of distributions on $ℝ$ generated by convolutions of Γ-distributions and the class generated by convolutions of mixtures of exponential distributions are generalized to higher dimensions and denoted by $T\left({ℝ}^d\right)$ and $B\left({ℝ}^d\right)$ . From the Lévy process $\left\{X_t^{\left(\mu \right)}\right\}$ on ${ℝ}^d$ with distribution μ at t=1, Υ(μ) is defined as the distribution of the stochastic integral ${\int }_0^{1}log(1/t)d{X}_t^{\left(\mu \right)}$ . This mapping is a generalization of the mapping Υ introduced by Barndorff-Nielsen and Thorbjørnsen in one dimension. It is proved that $\Upsilon \left(\mathrm{ID}\right({ℝ}^d\left)\right)=B\left({ℝ}^d\right)$ and $\Upsilon \left(L\right({ℝ}^d\left)\right)=T\left({ℝ}^d\right)$ , where $\mathrm{ID}\left({ℝ}^d\right)$ and $L\left({ℝ}^d\right)$ are the classes of infinitely divisible distributions and of self-decomposable distributions on ${ℝ}^d$ , respectively. The relations with the mapping Φ from μ to the distribution at each time of the stationary process of Ornstein-Uhlenbeck type with background driving Lévy process $\left\{X_t^{\left(\mu \right)}\right\}$ are studied. Developments of these results in the context of the nested sequence $L_m\left({ℝ}^d\right)$, $m=0,1,\dots ,\mathrm{\infty }$ , are presented. Other applications and examples are given.

Recursions for a class of sampling distributions of allele configurations are derived for the situation where the genealogy of the underlying population is modelled by a coalescent process with simultaneous multiple collisions of ancestral lineages. These recursions describe a new family of partition structures in terms of the composition probability function, parametrized by the infinitesimal rates of the coalescent process. For the Kingman coalescent process with only binary mergers of ancestral lines, the recursion reduces to that known for the classical Ewens sampling distribution. We solve the recursion for the star-shaped coalescent. The asymptotic behaviour of the number K_n of alleles (types) for large sample size n is studied, in particular for the star-shaped coalescent and the Bolthausen-Sznitman coalescent.

Using the concept of the convex hull of a set of lines, a dual random triangle is defined by selecting three lines from a parent triangle of lines. The angles of the constructed triangle define the shape; calculation of the shape distribution is described. For a sequence of nested triangles constructed in this way it is shown that there is convergence to collinearity and to the collinear shape distribution derived by Mannion for a sequence of vertex-generated triangles.

We consider a reversible R^d-valued Markov process {X_i; i≥0} with the unique invariant measure μ(dx)=f(x)dx, where the density f is unknown. The large-deviation principles for the nonparametric kernel density estimator f_n^* in L¹(R^d,dx) and for {||f_n^*-f||}₁ are established. This generalizes the known results in the independent and identically distributed case. Furthermore, we show that f_n^* is asymptotically efficient in the Bahadur sense for estimating the unknown density f.

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>½. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

We study a one-parameter family of attractive reversible nearest particle systems on a finite interval. As the length of the interval increases, the time at which the nearest particle system first hits the empty set increases from logarithmic to exponential depending on the intensity of interaction. In the critical case, the first hitting time is polynomial in the interval length.

For $T\ne 1$ , the domain G is T-homogeneous if TG=G. If $0\neg \in G$ , then necessarily $0\in \partial G$ . It is known that for some p>0, the Martin kernel K at infinity satisfies $K\left(\mathrm{Tx}\right)=T^{\mathrm{pK}}\left(x\right)$ for all $x\in G$ . We show that in dimension $d\ge 2$ , if G is also Lipschitz, then the exit time τ_G of Brownian motion from G satisfies $P_x\left({\tau }_G>t\right)\approx K\left(x\right)t^{-p/2}$ as $t\to \mathrm{\infty }$ . An analogous result holds for conditioned Brownian motion, but this time the decay power is $1-p-d/2$ . In two dimensions, we can relax the Lipschitz condition at 0 at the expense of making the rest of the boundary C².

In 2002 Basrak, Davis and Mikosch showed that an analogue of the Cramér-Wold device holds for regular variation of random vectors if the index of regular variation is not an integer. This characterization is of importance when studying stationary solutions to stochastic recurrence equations. In this paper we construct counterexamples showing that for integer-valued indices, regular variation of all linear combinations does not imply that the vector is regularly varying. The construction is based on unpublished notes by Harry Kesten.

Estimating the mean in a nonparametric regression on a two-dimensional regular grid of design points is asymptotically equivalent to estimating the drift of a continuous Gaussian process on the unit square. In particular, we provide a construction of a Brownian sheet process with a drift that is almost the mean function in the nonparametric regression. This can be used to apply estimation or testing procedures from the continuous process to the regression experiment as in Le~Cam's theory of equivalent experiments. Our result is motivated by first looking at the amount of information lost in binning the data in a density estimation problem.

The problem of prediction is considered in a multidimensional setting. Extending an idea presented by Barndorff-Nielsen and Cox, a predictive density for a multivariate random variable of interest is proposed. This density has the form of an estimative density plus a correction term. It gives simultaneous prediction regions with coverage error of smaller asymptotic order than the estimative density. A simulation study is also presented showing the magnitude of the improvement with respect to the estimative method.

We define a multivariate gamma distribution on $R^n$ by its Laplace transform ${\left(P(-\theta )\right)}^{-\lambda }$, $\lambda >0,$ where

$P\left(\theta \right)={\sum }_{T\subset \{1,\dots ,n\}}{p}_T{\prod }_{i\in T}{\theta }_i.$

Under $p_{\{1,\dots ,n\}}\ne 0$, we establish necessary and sufficient conditions on the coefficients of $P,$ such that the above function is the Laplace transform of some probability distribution, for all $\lambda >0,$ thus characterizing all infinitely divisible multivariate gamma distributions on $R^n.$

Bernoulli 11(5), 2005, 829-845. It has recently come to our attention that there is a large overlap between our Theorem 1.2 and the work of Tanaka (1973) and Cuesta and Matran (1989). In particular, the same result as ours is proved in both cases by analysing the same subadditive sequence, for real-valued random variables in Tanaka (1973) and in the more general Hilbert space setting in Cuesta and Matran (1989). Both these papers assume the stronger condition of a finite fourth moment, though they both comment that this can be weakened, perhaps to just requiring a finite (2 + δ)th moment, for δ > 0. In our paper we need only a finite second moment. We acknowledge the priority of the above authors on this result and sincerely apologize for our oversight.