Transcription of Discrete Stochastic Processes, Chapter 7: Random Walks ...
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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.
processes, which we have already studied and which will provide additional insight into the general study of random walks. After this, Sections 7.2 and 7.3 show how two major application areas, G/G/1 queues and
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