Two-Dimensional Recurrent Random Walk

Abstract

Just about all worthwhile known results concerning random walk (or concerning any stochastic process for that matter) are closely related to some stopping time T as defined in definition D3.3. Thus we plan to investigate stopping times. Given a stopping time T we shall usually be concerned with the random variable x _T., the position of the random walk at a random time which depends only on the past of the process. There can be no doubt that problems concerning x _T represent a natural generalization of the theory in Chapters I and II; for in those chapters our interest was confined to the iterates P _n (0,x) of the transition function—in other words, to the probability law governing x _n at an arbitrary but nonrandom time.