Random walks and random forests

Abstract

This chapter is inspired by the following quotation from Harris’s 1952 paper [193, §6]: Walks and trees. Random walks and branching processes are both objects of considerable interest in probability theory. We may consider a random walk as a probability measure on sequences of steps-that is, on “walks”. A branching process is a probability measure on “trees”. The purpose of the present section is to show that walks and trees are abstractly identical objects and to give probabilistic consequences of this correspondence. The identity referred to is nonprobabilistic and is quite distinct from the fact that a branching process, as a Markov process, may be considered in a certain sense to be a random walk, and also distinct from the fact that each step of the random walk, having two possible directions, represents a twofold branching.