extrahmwrk_m452f10

# extrahmwrk_m452f10 - Math 452 Extra Homework Take home as...

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Unformatted text preview: Math 452: Extra Homework. Take home as extra-homework. This is not a test and will not be marked but will possibly be discussed in class. EH1 Let X n be a Markov chain with transition probabilities p ij and consider the indicator function: I { X n = j } = braceleftbigg 1 , if X n = j , otherwise The long-run distribution is defined as π j = lim n −→∞ 1 n n summationdisplay m =1 I { X m = j } , with probability one, i.e, E [ π j ] = π j . (a) Show that π j = lim n −→∞ 1 n n summationdisplay m =1 P ( m ) ij . Hint: Show that E [ I { X n = j } ] = P { X n = j } . (b) Let N n be a random time uniformly distributed on { 1 , 2 , ··· ,n } ,i.e., P { N n = i } = 1 /n, i = 1 , 2 , ··· ,n . Show that lim n −→∞ P { X N n = j/X = i } = lim n −→∞ P { X N n = j } = π j , ∀ i,j = 0 , 1 , 2 , ··· , (c) Let T jj be the random time it takes for the Markov chain to return to state j (also known as the first passage time): T jj = min { n ≥ 1 : X n = j/X = j } and let...
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extrahmwrk_m452f10 - Math 452 Extra Homework Take home as...

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