# ass3 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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STAT455/855 Fall 2009 Applied Stochastic Processes Assignment #3 Due Monday, Nov.9 Starred questions are for 855 students only. 1. Let { X n : n 0 } be an irreducible, aperiodic, positive recurrent Markov chain, with stationary distribution π . (a) Show that P ( X n = j ) 1 j as n → ∞ , where μ j is the mean return time to state j . (b) Show that if { x n } n =1 is a sequence of real numbers satisfying x n x as n → ∞ for some limit x ( -∞ , ), then n - 1 n i =1 x i x . Hence show that π j = lim n →∞ 1 n n X m =1 P ( X m = j ) (so we often interpret π j to be the long run proportion of time that the chain is in state j ). 2. Let X = { X n : n 0 } be and irreducible Markov chain, Y = { Y n : n 0 } an independent copy of X , and let Z = { Z n : n 0 } , where Z n = ( X n , Y n ). (a) If X has period d > 1 show that Z cannot be irreducible. (b) If

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ass3 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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