Random Measures with Aftereffects

Abstract

A class of $\mathscr{D}$ of random measures, generalizing the class of completely random measures, is developed and shown to contain the class of Poisson cluster point processes. An integral representation is obtained for $\mathscr{D}$, generalizing the Levy-Ito representation for processes with independent increments. A subclass $\mathscr{D}_n \subset \mathscr{D}$ is defined such that for $X \in \mathscr{D}_n$, the distribution of the random vector $X(A_1), \cdots, X(A_m), m > n, A_1, \cdots, A_m$ disjoint, is determined by the distributions of all subvectors $X(A_{i_1}), \cdots, X(A_{i_k}), 1 \leqq k \leqq n$. The class $\mathscr{D}_1$ coincides with the class of completely random measures.