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

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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 π such that: – πP = π , meaning ∑ i π i P ( i, j ) ∀ j ∈ S – This π 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 ) = π j for all j ∈ S . • Idea of most proofs: – Show that π exists – Show lim n →∞ P n ( i, j ) = π j – Uniqueness of π is easy. • Explicit representations of π – My favorite (not in text): let T i = inf { n : n > 1 , X n = i } where inf φ = ∞ . – Then E i (# of visits to j before T i ) = π j π i where π is the unique invariant probability distribution –...
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lecture17 - Lecture 17 Long run behaviour of Markov chains...

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