Random Walks and Martingales

Let {Xi; i ≥ 1} be a sequence of IID random variables, and let Sn=X1+X2+...+Xn. The integer time stochastic process {Sn;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, {Sn;n≥1}, and thus in such questions as finding the first n for which Snexceeds some threshold a, or the probability that Snexceeds a for any value of n. Since Sndrifts downward with increasing n for \(E[X] = \bar{X} 0\), the results to be obtained depend critically on whether \(\bar{X} 0\) or \(\bar{X} = 0\). Since results for \(\bar{X} 0\) by considering {−Sn;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.