hw 5 - E i T i = 1/π i(c Use(a to prove E i T i 2 = 2 E i...

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3. Let P ( i,j ) be a Markov transition matrix on { 0 , 1 , 2 ,... } . Give a simple necessary and suﬃcient condition, in terms of P , for the following assertion to be true. For any pair i 0 < j 0 it is possible to construct ( X n ,Y n ; n 0) such that 1. X is the ( i 0 ,P )-chain 2. Y is the ( j 0 ,P )-chain 3. X n Y n for all n . 4. Let ( X n ) be irreducible positive-recurrent with stationary distribution π . Fix a subset B of S . Let T B = min { n 1 : X n B } . A kn = { X m B c for all k m n } . (a) Show that for the stationary chain, P ( A kn ) depends only on n - k , and deduce that for the stationary chain P ( X 0 B,T B n ) = P ( T B = n ) . (b) Use (a) to give a new proof that
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Unformatted text preview: E i T i = 1 /π ( i ). (c) Use (a) to prove E i ( T i ) 2 = 2 E i T i ( X j ( E j T i /E j T j )-1) . 5. Let ( X n ; n ≥ 0) be a ﬁnite-state irreducible Markov chain. Write π for the stationary distribution and T j = min { n ≥ 0 : X n = j } for the ﬁrst hitting time. (a) Prove that ∑ j π j E i T j does not depend on i . (b) Give an example to show that ∑ i π i E i T j may depend on j . 5...
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