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Unformatted text preview: Stat 333 Equilibrium Distributions and the Renewal Theorem Let X n be a discrete-time Markov chain with state space S and transition matrix P . Let the row vector p n denote the marginal distribution of the chain at step n , i.e., p n ( j ) = P ( X n = j ) for j S . Note that p n ( j ) 0 and j S p n ( j ) = 1. Thus p n is a probability vector . Theorem (the updating rule): p n +1 = p n P Suppose C is a closed class with submatrix P C . A probability vector is called an equilibrium distribution for the class if it satisfies = P C subject to summationdisplay j C j = 1 (*) Thus an equilibrium distribution remains unchanged under updating. (note: we use the notation j , rather than ( j ), for convenience, because there is no need to indicate the time n of transition here.) may be determined either by solving the system of linear equations given by (*) or by guessing at the solution and then verifying that (*) holds. Main Applied Result: Let C be a class (periodic or aperiodic) of a Markov chain. Then (a) if C consists of positive recurrent states, then C has a unique equilibrium distribution ....
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This note was uploaded on 07/12/2010 for the course STAT 333 taught by Professor Chisholm during the Winter '08 term at Waterloo.
- Winter '08