4. For each of the below listed descriptions, provide the transition graph of a Markov Chain that satisfies them. Explain in a few sentences why your example has the properties. If such an example does not exist, explain why: (a) A 2 state Markov Chain without limiting distribution but with at least one stationary distribution. (b) A 2 state Markov Chain with limiting distribution. Determine it. (c) An irreducible 4 state Markov Chain which is not aperiodic. (d) An irreducible 3 state Markov Chain which is aperiodic. (e) An irreducible 4 state Markov Chain which is aperiodic and such that the expected return time to state 1 is infinite (i.e. E[T₁|X0 = 1] = ∞.). (f) A Markov Chain with more than one communication class and a limiting distribution.

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