In this paper we investigate the cluster behavior of linearly interacting Brownian motions indexed by . We show that (on a logarithmic scale) the block average process converges in path space to Brownian motion.

We construct a catalytic super process X (measure-valued spatial branching process) where the local branching rate is governed by an additive functional A of the motion process. These processes have been investigated before but under restrictive assumptions on A. Here we do not even need...

Classical super-Brownian motion (SBM) is known to take values in the space of absolutely continuous measures only if d=1. For d[greater-or-equal, slanted]2 its values are almost surely singular with respect to Lebesgue measure. This result has been generalized to more general motion laws and...

A strong law of large numbers is presented for a class of random variables X0, X1,..., which satisfy for a suitable function [latin small letter f with hook](x) > x.

We study asymptotic properties of non-negative random variables Xn, n[greater-or-equal, slanted]0, satisfying the recursion . If the functions g(x) and [sigma]2(x) are properly balanced at infinity, Xn is asymptotically [Gamma]-distributed in a suitable scale. This result contains several known...

We generalize a result by Kozlov on large deviations of branching processes (Zn) in an i.i.d. random environment. Under the assumption that the offspring distributions have geometrically bounded tails and mild regularity of the associated random walk S, the asymptotics of is (on logarithmic...

We establish a multivariate empirical process central limit theorem for stationary -valued stochastic processes (Xi)i=1 under very weak conditions concerning the dependence structure of the process. As an application, we can prove the empirical process CLT for ergodic torus automorphisms. Our...

The positive dependence notion of association for collections of random variables is generalized to that of weak association for collections of vector valued random elements in such a way as to allow negative dependencies in individual random elements. An invariance principle is stated and...

Associated sequences have been studied extensively in recent years. Burton et al. (1986) introduced a generalization of association to d-dimensional random vectors and proved Functional Central Limit Theorems. In the present paper a Berry-Esséen theorem and a Functional Law of the Iterated...

We present a new technique for proving the empirical process invariance principle for stationary processes (Xn)n=0. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions (f(Xn))n=0, not containing...