We obtain some liminf limits for the Wiener sheet. The approach relies on a careful analysis of the lower tail of the Ornstein-Uhlenbeck process. Our results can be applied to normalized Kiefer and empirical processes. In particular, they yield a satisfying answer to Hirsch's problem for the...

We show that the occupation measure on the path of a planar Brownian motion run for an arbitrary finite time interval has an average density of order three with respect to the gauge function . In other words, almost surely,We also prove a refinement of this statement: Almost surely, at -almost...

We give necessary and sufficient conditions under which a symmetric measurable infinitely divisible process has sample paths in an Orlicz space L[psi] with a function [psi] satisfying the [Delta]2 condition and, as an application, obtain necessary and sufficient conditions for a symmetric...

A general procedure, inspired from that used for deterministic partial differential equations, is presented to reduce the Zakai stochastic Pde of filtering on n to a stochastic Pde on a lower-dimensional space m, with m n. The method is based upon invariance group techniques. We show how the...

By using logarithmic transformations and stochastic analysis, an explicit lower bound of Dirichlet heat kernels is obtained, which can be sharp for both short time and long time. Next, a two-side comparison theorem is presented for Dirichlet heat kernels and some closed ones, from which we...

This paper concerns a partially observable finite horizon control problem for -valued pure Markov jump process using the information given by the point process which counts the total number of jumps. Equivalence between the partially observable control problem and the separated control problem...

We consider some classes of stationary, counting-measure-valued Markov processes and their companions under time reversal. Examples arise in the Lévy-Itô decomposition of stable Ornstein-Uhlenbeck processes, the large-time asymptotics of the standard additive coalescent, and extreme value...

We study the weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions involving majorizing measures. As an application, we consider the weak convergence of stochastic processes of the form where {Xj}j=1[infinity] is a sequence of i.i.d.r.v.s with...

We prove that, under appropriate conditions, the sequence of approximate solutions constructed according to the Euler scheme converges weakly to the (unique) solution of a stochastic differential equation with discontinuous coefficients. We also obtain a sufficient condition for the existence of...

We consider a classical risk process compounded by another independent process. Both of these component processes are assumed to be Lévy processes. Sharp conditions are given on the parameters of these two components to ensure when ruin is certain, and also when the time of ruin is of...