Random Walks and Martingales

Abstract

Let {X_i; i ≥ 1} be a sequence of IID random variables, and let S_n=X₁+X₂+...+X_n. The integer time stochastic process {S_n;n≥1} is called arandom walk.For any given n, S, is just a sum of IID random variables, but here, we are more interested in the behavior of the random walkprocess, {S_n;n≥1}, and thus in such questions as finding the first n for which S_nexceeds some threshold a, or the probability that S_nexceeds a for any value of n. Since S_ndrifts downward with increasing n for $E[X] = \bar{X} < 0$ and S_n drifts upward if $\bar{X} > 0$, the results to be obtained depend critically on whether $\bar{X} < 0,\bar{X} > 0$ or $\bar{X} = 0$. Since results for $\bar{X} < 0$ can be easily translated into results for $\bar{X} > 0$ by considering {−S_n;n≥0}, we will focus on the case $\bar{X} < 0$. As one might expect, both the results and the techniques have a very different flavor when $\bar{X} = 0$, since here the random walk does not drift but typically wanders around in a rather aimless fashion. We first give several representative examples of random walks and then treat the problem of threshold crossings. We then introduce a rather general type of stochastic process called a Martingale. The topic of Martingales is both a subject of interest in its own right and also a tool that provides additional insight into random walks, laws of large numbers, and other basic topics in probability and stochastic processes.