Question 3. Recall that the discrete stochastic process {Xn, n = 0, 1, 2,...} is a Markov chain if for each n, P(Xn+1 = j|Xn = i, Xn-1 = in-1,..., Xo = io) = P(Xn+1 = j|Xn = i) = Pij. Let

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Question 3. Recall that the discrete stochastic process {Xn, n = 0, 1, 2,...} is a Markov chain if for each n, P(Xn+1 = j|Xn = i, Xn-1 = in-1,..., Xo = io) = P(Xn+1 = j|Xn = i) = Pij. Let {Xn, n = 0, 1, 2,...} be a Markov chain, does the following hold as well? P(Xn+2=jXn = i, Xn-1 = in-1, ..., Xo = io) = P(Xn+2 = j|Xn = i) Give a proof if you think it is true, otherwise give a counterexample. Hint: You may use the law of total probability for conditional probability without proof.

 

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