Stationary point processes by Markov-dependent intervals and infinite intensity

Abstract

Null-recurrent random walks with symmetric step distribution are mapped to give null-recurrent Markov chains on the interval (0, 1). These chains are used to construct stationary Wold processes (point processes with Markov-dependent intervals) with infinite intensity, which processes are simple (i.e., almost surely orderly) but not (analytically) orderly. As examples of point processes with infinite intensity, they are constructed without any intermediate step that relies on using point processes of finite intensity.