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Section 2 - Notes for Section Feb 8-12 ORIE 3510 Review...

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Notes for Section Feb 8-12 ORIE 3510 Review accessibility, communication, decomposition of state space, irreducibil- ity transience, recurrence, positive recurrence, periodicity, ergodicity as solidarity (class) properties limiting probabilities, stationary probabilities, long-run proportions Problem 14 P 1 : { 1 , 2 , 3 } irreducible recurrent P 2 : { 1 , 2 , 3 , 4 } irreducible recurrent P 3 : { 1 , 3 } , { 4 , 5 } recurrent closed; { 2 } transient P 4 : { 1 , 2 } , { 3 } recurrent closed; { 4 } , { 5 } transient Problem 15 Suppose the state space of a Markov chain has M states and i j . Then P ( n ) ij > 0 for some n , so we can find a path leading from i to j with positive probability, i.e. P ( X n = j, X n - 1 = i n - 1 , . . . , X 1 = i 1 | X 0 = i ) > 0 for some sequence of states i 1 , . . . , i n - 1 . We can write this probability as P ( X n = j | X n - 1 = i n - 1 ) . . . P ( X 1 = i 1 | X 0 = i ) = P i,i 1 P i 1 ,i 2 . . . P i n - 1 ,j 1
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by conditioning and applying the Markov property. Furthermore, we can assume that the path visits each state at most once. In particular, suppose that the sequence is i 1 , . . . , i k , i k +1 , . . . , i k 0 , i k 0 +1 , . . . , i n and
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