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 longrun 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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 Spring '08
 Chisholm
 Probability theory, Markov chain, Random walk

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