Random walk point processes

Abstract

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.