Random Walks

Abstract

Random walks entered mathematics early on through the analysis of gambling and other games of chance. To cite a typical example, let X ₀ denote the initial fortune of a certain gambler and let X _n stand for the amount won (if X _n ≥ 0) or lost (if X _n ≤ 0) the nth time that the gambler places a bet. In the simplest gambling situations, the X _n’s are i.i.d., and the gambler’s fortune at time n is described by the partial sum $Sn = \sum\nolimits_{j = 0}^n {{X_j}}$. The stochastic process S = (S _n ; n ≥ 0) is called a one-dimensional random walk and lies at the heart of modern, as well as classical, probability theory. This chapter is a study of some properties of systems of such walks.

Those cannot remember the past are condemned to repeat it.

—Santayanna