Ergodicity and inequalities in a class of point processes

Abstract

In this paper we extend some results from renewal theory to a certain class of point processes. Firstly, a strong version of Blackwell's renewal theorem is shown to hold when the memory of the process at time t, say, contains only the configuration of the latest m points of occurrence preceding t, and of the points in the interval [ t− A, t], where A is a constant. Secondly, a generalization of the decreasing failure rate (DFR) concept is introduced, based on the following principle: “if there have been many points of occurrence recently, then we will soon experience another one”. Inequalities and monotonicity results are established under this new type of DFR assumption.