Discrete Stochastic Processes, Chapter 7: Random Walks ...

Chapter 7 Random Walks , LARGE DEVIATIONS, AND MARTINGALES introduction Definition 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 a Random walk, or, more precisely, the one-dimensional Random walk based on {Xi; i 1}. For any given n, Sn is simply a sum of IID Random variables, but here the behavior of the entire Random walk process, {Sn; n 1}, is of interest. Thus, for a given real number > 0, we might want to find the probability that the sequence {Sn; n 1} contains any term for which Sn ( , that a threshold at is crossed) or to find the distribution of the smallest n for which Sn . We know that Sn/n essentially tends to E [X] = X as n 1.

7.1 Introduction Definition 7.1.1. Let {X i; i ≥ 1} be a sequence of IID random variables, and let S n = X 1 + X 2 + ··· + X n. The integer-time stochastic process {S n; n ≥ 1} is called a random walk, or, more precisely, the one-dimensional random walk based on {X i; i ≥ 1}. For any given n, S n is simply a sum of IID random ...

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