A Martingale Approach to the Poisson Convergence of Simple Point Processes

Abstract

The paper concerns the Doob-Meyer increasing processes of simple point processes on the positive half line. It is shown that the weak convergence of such point processes to a simple Poisson process is implied by the pointwise weak convergence of their increasing processes, provided that the increasing processes satisfy a mild regularity condition. Conditions under which the regularity is satisfied are investigated. One condition is that the increasing process is that of the point process with its generated $\sigma$-fields. The Poisson convergence theorem is applied to superpositions of point processes.