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. Thus if X< 0, Sn will tend to drift downward and if X > 0, Sn will tend to drift upward. This means that the results to be obtained depend critically on whether X< 0, X> 0, or X = 0.

takes on the values 1 and 0, whereas for a simple random walk X takes on the values 1 and -1. ... in which a subsequence of values s n approach k as a limit but never quite reach k. This is impossible for a simple random walk since all s ... threshold at any given positive value ...

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  Value, Limits, Thresholds

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