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# Lec17 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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Stat 150 Stochastic Processes Spring 2009 Lecture 17: Limit distributions for Markov Chains Lecturer: Jim Pitman Finite state space Markov chain X 0 , X 1 , . . . with state space S having N ele- ments, N < . Matrix P , P i ( X n = j ) = P n ( i, j ). Problem : Suppose you know the initial distribution λ of X 0 , λ j = P ( X 0 = j ). Want to evaluate lim n →∞ P λ ( X n = j ). Notation: P λ ( · ) := i λ i P i ( · ) Questions : When does limit F exist? If F exists, how to evaluate it? We say P is regular if n : P n ( i, j ) > 0 for all i, j S . Equivalently , there exists for every i and j some sequence i 0 = i, i 1 , i 2 , . . . , i n = j with P ( i k , i k +1 ) > 0 for all 1 k n some path from i to j in exactly n steps. Obviously P n ( i, j ) > 0 , i, j = P m ( i, j ) > 0 , i, j, m n : Let m = n + k , P n + k ( i, j ) = X l P k ( i, l ) P n ( l, j ) > 0 because P n ( l, j ) > 0 for all n and l P k ( i, l ) = 1 P k ( i, l ) > 0 for some l . Theorem: If P is a regular transition matrix on a finite set, then there ex- ists a unique probability distribution π on S such that πP = π ( π is called the

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Lec17 - Stat 150 Stochastic Processes Spring 2009 Lecture...

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