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RandomWalkOnACircle - STAT 333 Random Walk On A Circle The...

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STAT 333 Random Walk On A Circle The ordinary random walk (which we have studied in great detail) can be modified in various ways by the introduction of different kinds of boundaries. The introduction of boundaries typically changes the long-run behaviour of the process in some significant way. For example, the gambler’s ruin can be viewed as a random walk in which we have placed absorbing boundaries at 0 and k . The process eventually hits one of the boundaries and stops (unlike the ordinary walk which keeps bouncing around forever). A different type of example occurs when we place a reflecting boundary at 0. Here, whenever the process hits 0, the process bounces (or reflects) back to state 1 at the next step. This example is described in Question #6 on Assignment 2. In this question you are asked to determine the behaviour of “return to 0” in the reflecting walk – the results are not the same as that of the ordinary random walk, but, if you are careful, you can use results from the ordinary random walk to solve the reflecting walk in just a few lines. The Circular Random Walk Another way of introducing a boundary into the ordinary random walk is to fix an integer m 2 and make the identification m 0, then carry out arithmetic mod m . This is equivalent to taking
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