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Consider a Markov chain {Xn; n = 0, 1, 2, . . . } on S...

Consider a Markov chain {Xn; n = 0, 1, 2, . . . } on S = N = {0, 1, 2, . . . } with transition probabilities P(x, 0) = 1/2 , P(x, x + 1) = 1/2 ∀x ∈ S, .

(a) Show that the chain is irreducible.

(b) Find P0(T0 = n) for each n = 1, 2, . . . .

(c) Use part (b) to show that state 0 is recurrent; i.e., ρ00 = 1.

(d) Use part (c) to show that state 0 is positive recurrent; i.e., m0 < ∞.

(e) With justification, does the chain have a stationary distribution? What is it?

(f) Is the process reversible? Justify your answer.

Math   Statistics-And-Probability   
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