RandomWalk_Summary

RandomWalk_Summary - STAT 333 Random Walk Summary The...

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STAT 333 Random Walk Summary The random walk is a Markov chain taking values in the integers S = { . . . , - 2 , - 1 , 0 , 1 , 2 , . . . } . We assume that the walk starts at the origin 0 (that is, X 0 = 0) although we may relabel the axis to make any chosen point the origin. On each step the process either jumps one unit to the right or one unit to the left. Therefore, we either have X n +1 = X n + 1 or X n +1 = X n - 1 where X n = position of the walk after n steps. The jumps themselves (+ or -) can be viewed as a sequence of Bernoulli trials (or coin tosses) where p is the probability of a jump to the right (+) and q = 1 - p is the probability of a jump to the left (-). We assume that 0 < p < 1. The event λ 0 0 = “return to 0” is a renewal event of period d = 2. Let T 0 0 = number of steps to first return to 0. Then the probability of at least one return to 0 is f 0 0 = P ( T 0 0 < ) = 1 - | p - q | . Therefore λ 0 0 is transient if p negationslash = 1 / 2 and recurrent if p = 1 / 2. We further calculate E ( T 0 0 ) = when p = 1 / 2 so “return to 0” is null recurrent for the balanced walk. Let V 0 0 = number of returns to 0. Then E ( V 0 0 ) = f 0 0 1 - f 0 0 = 1 - | p - q | | p - q | Let k be a positive integer and let
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