lecture17 - Lecture 17 : Long run behaviour of Markov...

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Unformatted text preview: Lecture 17 : Long run behaviour of Markov chains STAT 150 Spring 2006 Lecturer: Jim Pitman Scribe: Vincent Gee <> Basic Case: S is finite Markov matrix is P Assume that for some power of P has all entries > 0: k such that P k ( i, j ) > i, j S Such P is called regular Then (Theorem): a unique stationary probability distribution such that: P = , meaning i i P ( i, j ) j S This is then the limit distribution of X n as n , no matter what the initial distribution of X i.e. lim n P n ( i, j ) = lim n P i ( X n = j ) = j for all j S . Idea of most proofs: Show that exists Show lim n P n ( i, j ) = j Uniqueness of is easy. Explicit representations of My favorite (not in text): let T i = inf { n : n > 1 , X n = i } where inf = . Then E i (# of visits to j before T i ) = j i where is the unique invariant probability distribution...
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lecture17 - Lecture 17 : Long run behaviour of Markov...

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