# ass2 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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STAT455/855 Fall 2009 Applied Stochastic Processes Assignment #2 Due Friday, Oct.16 Starred questions are for 855 students only. 1. (a) Let { X n : n 0 } be an irreducible Markov chain with period d > 1 and transition matrix P . Consider the Markov chain with transition matrix P k , where k { 1 ,...,d } . (i) For each k , give the equivalence classes of this Markov chain and the period of each state. (ii) (bonus marks) Prove your assertions. (b) Let { X n : n 0 } be a Markov chain and let Z 1 ,Z 2 ,... be a sequence of i.i.d. random variables on the positive integers, each with probability mass function p Z . Let T 0 = 0, T n = n i =1 Z i , and Y n = X T n for n 0. Is { Y n : n 0 } a Markov chain? If yes, ﬁnd its transition probability matrix. (c) Give an example of an irreducible Markov chain with period d > 1 such that the cyclic classes are not all the same size. 2. Consider a Markov chain on the set S = { 0 , 1 , 2 ,... } with transition probabilities p i,i +1 = a i , i 0, p i, 0 = 1 - a i , where { a i ,i 0 } is a sequence of constants which satisfy 0 < a i < 1 for all i . Let b 0 = 1, b i = a 0 a 1 ··· a i - 1 for i 1. (a) Show that the chain is recurrent if and only if b i 0 as i → ∞ and show that the chain is positive recurrent if and only if i =0 b i < . (Hint: First show that the chain is irreducible and then focus only on state 0. Let T 0 be the ﬁrst time the chain returns to state 0, starting in state 0. The chain is recurrent

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## This note was uploaded on 09/15/2010 for the course STAT 455/855 taught by Professor Glentakahara during the Fall '09 term at Queens University.

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ass2 - STAT455/855 Fall 2009 Applied Stochastic Processes...

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