4.3.2 Show that a finite-state aperiodic irreducible Markov chain is regular and recurrent.
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- Explain how you can determine the steady state matrix X of an absorbing Markov chain by inspection.5 Consider the Markov chain with transition matrix 1/2 (1/4 3/4) P = Find the fundamental matrix Z for this chain. Compute the mean first passage matrix using Z.Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3. Find limn→∞ P(Xn = 1).
- The following Markov chain is irreducible /1/3 0 0 0 1 2/3 0 1/5 4/5/Let X be a Markov chain and let (nrr≥ 0} be an unbounded increasing sequence of positive integers. Show that Yr Xnr constitutes a (possibly inhomogeneous) Markov chain. Find the transition matrix of Y when nr = 2r and X is: (a) simple random walk, and (b) a branching process. =Consider a continuous-time Markov chain whose jump chain is a random walk with reflecting barriers 0 and m where po,1 = 1 and pm,m-1 =1 and pii-1 = Pii+1 = for 1
