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0 and show that it does not satisfy the detailed balance condition (1.11).
(b) Consider
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a
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1a
21b
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b
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3
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1c
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c
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d
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1d
0
and show that there is a stationary distribution satisfying (1.11) if
0 < abcd = (1 a)(1 b)(1 c)(1 d). 1.13. Consider the Markov chain with transition matrix:
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2
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4 1
0
0
0.8
0.4 2
0
0
0.2
0.6 3
0.1
0.6
0
0 4
0.9
0.4
0
0 (a) Compute p2 . (b) Find the stationary distributions of p and all of the
stationary distributions of p2 . (c) Find the limit of p2n (x, x) as n ! 1.
1.14. Do the following Markov chains converge to equilibrium?
(a) 1
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1 2
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.7
0 3
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.5
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0 4
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.5
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1 .0
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0 65 1.12. EXERCISES
(c) 1
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0 3
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0 4
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1
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0 5
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.8 6
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.6
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0 1.15. Find limn!1 pn (i, j ) for
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2
p=
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1/3 You are supposed to do this and the next problem by solving...
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 Spring '10
 DURRETT
 The Land

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