We study properties of stable-like laws, which are solutions of the distributional equation where (N,A1,A2,...) is a given random variable with values in {0,1,...}x[0,[infinity])x[0,[infinity])x..., and Z,Z1,Z2,... are identically distributed positive random variables, independent of each other...

The class of haploid population models with non-overlapping generations and fixed population size N is considered such that the family sizes [nu]1,...,[nu]N within a generation are exchangeable random variables. A criterion for weak convergence in the Skorohod sense is established for a properly...

We obtain a strong approximation for the logarithmic average of sample extremes. The central limit theorem and laws of the iterated logarithm are immediate consequences.

In this paper we find the Wiener chaos expansion for the local time of the fractional Brownian motion with Hurst parameter H and we derive a Tanaka formula in the case . As an application we deduce an Itô's formula for convex functions.

Given an extremal-[Lambda] process {Y[Lambda](t), t0}, the transformed process {U(s)=Y[Lambda](es)-s, -[infinity]s[infinity]} is a stationary strong Markov process. We prove an almost sure invariance principle for the process . By an approximation this yields an almost sure invariance principle...

Kella and Whitt (J. Appl. Probab. 29 (1992) 396) introduced a martingale {Mt} for processes of the form Zt=Xt+Yt where {Xt} is a Lévy process and Yt satisfies certain regularity conditions. In particular, this provides a martingale for the case where Yt=Lt where Lt is the local time at zero of...

The paper considers the top Lyapunov exponent of a two-dimensional linear stochastic differential equation. The matrix coefficients are assumed to be functions of an independent recurrent Markov process, and the system is a small perturbation of a nilpotent system. The main result gives the...

We study a discrete time interacting particle system which can be considered as an annihilating branching process on where at each time every particle either performs a jump as a nearest neighbor random walk, or splits (with probability [var epsilon]) into two particles which will occupy the...

We consider a linear sequence of 'nodes', each of which can be in state 0 ('off') or 1 ('on'). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are 'on'. These nodes are immediately switched off, and become on again...

A fractional version of the heat equation, involving fractional powers of the negative Laplacian operator, with random initial conditions of exponential type, is introduced. Two cases are considered, depending on whether the Hopf-Cole transformation of such random initial conditions coincides,...