Transcription of 15 Markov Chains: Limiting Probabilities
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15 Markov Chains: Limiting Probabilities
15 Markov CHAINS: Limiting PROBABILITIES16715 Markov Chains: Limiting ProbabilitiesExample that the transition matrix is given byP= 00 1 0 .Recall that then-step transition Probabilities are given by powers ofP. So let s look at somelarge powers ofP, beginning withP4= .Then, to four decimal placesP8 .and subsequent powers are the same to this matrix elements appear to converge and the rows become almost identical. Why? Whatdetermines the limit? These questions will be answered in this say that a statei Shasperiodd 1 if (1)Pnii>0 implies thatd|n, and (2)dis thelargest positive integer that satisfies (1).Example random walk onZ, withp (0,1).
15 MARKOV CHAINS: LIMITING PROBABILITIES 170 This is an irreducible chain, with invariant distribution π0 = π1 = π2 = 1 3 (as it is very easy to check). Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. Although the chain does spend 1/3 of the time at each state, the transition
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