The Simple Random Walk I: Associated Boundary Value Distributions, Transience, and Recurrence

The simple random walk is the generic example of a discrete time temporal evolution on an integer state space. In this chapter it is defined and simple combinatorics are provided in the computation of its distribution. Two possible characteristic long-time properties, (point) recurrence and transience, are identified in the course of the analysis. Recurrence is a form of “stochastic periodicity” in which the process revisits a state (or arbitrarily small neighborhood) infinitely often, while transience refers to the phenomena in which there are at most finitely many returns.