2. Consider the Markov chain X = (Xn)neN with state space I = {A, B, C, D, E, F, G, H} and one step transition probabilities given in the following diagram: B TH HIN TH 3 4 E 11 13 1 2137 (a) Decompose the state space into its communicating classes and state the period of each class. Hence, identify the set of transient states T and a communicating class of recurrent states R. (b) Write down the one-step transition matrix P for the discrete parameter Markov chain Y with state space R, that is, the restriction of the Markov chain X to the recurrent class RCI. (c) What conditions does an invariant probability mass function for a discrete time Markov chain satisfy? Find π for the Markov chain Y.

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