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hwk2 - E3106 Solutions to Homework 2 Columbia University...

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E3106, Solutions to Homework 2 Columbia University Exercise 13 . Proof For n > r and i, j , according to Chapman-Kolmogorov equations, we have P n ij = X k P n r ik P r kj . Since X k P n r ik = 1 , it follows that there exists some state k 0 such that P n r ik 0 > 0. And because P r k 0 j > 0, it follows that P n ij P n r ik 0 P r k 0 j > 0 , which completes the proof. Exercise 14 . (1) The classes of the states of the Markov chain with transition probability P 1 is { 0 , 1 , 2 } . Because it is a fi nite-state Markov chain, all the states are recurrent. (2)The classes of the states of the Markov chain with transition probability P 2 is { 0 , 1 , 2 , 3 } . And because it is a fi nite-state Markov chain, all the states are recurrent. (3)The classes of the states of the Markov chain with transition probability P 3 is { 0 , 2 } , { 1 } , { 3 , 4 } . It can be easily seen from a graph that { 0 , 2 } and { 3 , 4 } are recurrent, while { 1 } is transient. Here we also give a rigorous proof of this claim. (3.1) States 0,2 are recurrent. Firstly, we prove by induction that for all n , P n 00 = 1 2 and P n 02 = 1 2 . Obviously the result is true for n = 1. Then suppose P n 1 00 = 1 2 and P n 1 02 = 1 2 are true, it follows that P n 00 = P n 1 00 P 00 + P n
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