2 for concreteness suppose there are three balls in

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Unformatted text preview: ctors cannot possibly be perpendicular, which is a contradiction. 1.5 Limit Behavior P1 If y is a transient state, then by Lemma 1.11, n=1 pn (x, y ) < 1 for any initial state x and hence pn (x, y ) ! 0 This means that we can restrict our attention to recurrent states and in view of the decomposition theorem, Theorem 1.8, to chains that consist of a single irreducible class of recurrent states. Our first example shows one problem that can prevent the convergence of pn (x, y ). Example 1.21. Ehrenfest chain (continuation of 1.2). For concreteness, suppose there are three balls. In this case the transition probability is 0 1 2 3 0 1 2 3 0 3 /3 0 0 1/3 0 2 /3 0 0 2/3 0 1/3 0 0 3 /3 0 23 1.5. LIMIT BEHAVIOR In the second power of p the zero pattern is shifted: 0 1 2 3 0 1/3 0 2/9 0 1 2 3 0 2/3 0 7/9 0 2 /9 0 7/9 0 2/3 0 1 /3 To see that the zeros will persist, note that if we have an odd number of balls in the left urn, then no matter whether we add or subtract one the result will be an even number. Likewise, if the num...
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