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Unformatted text preview: oretical possibility, it rarely manifests itself in applications, except occasionally as an oddeven parity problem, e.g., the Ehrenfest chain. In most cases we will ﬁnd (or design) our chain to be aperiodic,
i.e., all states have period 1. To be able to verify this property for examples,
we need to discuss some theory.
Lemma 1.17. If p(x, x) > 0, then x has period 1.
Proof. If p(x, x) > 0, then 1 2 Ix , so the greatest common divisor is 1.
This is enough to show that all states in the weather chain (Example 1.3),
social mobility (Example 1.4), and brand preference chain (Example 1.5) are
aperiodic. For states with zeros on the diagonal the next result is useful.
Lemma 1.18. If ⇢xy > 0 and ⇢yx > 0 then x and y have the same period.
Why is this true? The short answer is that if the two states have di↵erent
periods, then by going from x to y , from y to y in the various possible ways,
and then from y to x, we will get a contradiction.
Proof. Suppose that the period of x is c, w...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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