If X and X1, X2,... are exchangeable processes on [0, 1] orR+, such that for all t, then , provided that X is continuous or ergodic and satisfies certain moment conditions. It is in fact enough to assume convergence for t restricted to some suitable subset. The proof uses analytic properties of...

In earlier work by the author, the convergence in distribution of a sequence of point processes towards a simple limit has been characterized by three conditions, where the first ensures convergence of the supports, the second that the limit of every convergent subsequence is simple, and the...

Given a process X on or , we may form a random sequence [xi]1,[xi]2,... by sampling from X at some independent points [tau]1,[tau]2,... . If X is stationary up to shifts (which holds for broad classes of Markov and Palm processes) and the distribution of ([tau]n) is asymptotically invariant (as...

From the predictable reduction of a marked point process to Poisson, we derive a similar reduction theorem for purely discontinuous martingales to processes with independent increments. Both results are then used to examine the existence of stochastic integrals with respect to stable Lévy...

By a symmetric interval partition we mean a perfect, closed random set [Xi] in [0,1] of Lebesgue measure 0, such that the lengths of the connected components of [Xi]c occur in random order. Such sets are analogous to the regenerative sets on , and in particular there is a natural way to define a...

Improving and extending a characterization of Poisson processes by Rényi, we present several mild conditions ensuring a point process to be of Poisson or sample type, and we give related conditions for convergence in distribution towards such processes. The basic property underlying our results...