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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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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