A Note on Poisson-Subordination

Abstract

Pseudo-Poisson processes can be obtained from discrete time Markov processes by subordination. A continuous time analogue of a random walk is defined by $Y(t) = S\lbrack T(t)\rbrack$ where $S(n)$ is the partial sum of a sequence of independent identically distributed random variables and $T(t)$ a process with stationary independent increments, independent of $S(n)$ and taking values in the non-negative integers. It is then shown that $Y(t)$ is a compound Poisson process; furthermore the supremum of $Y(t)$ is Poisson-subordinated to the maximum of $S(n)$ if and only if $T(t)$ is a Poisson process.