Solution: Consult iTunes course for full detailed solutions.
By the definition of a Markov chain, we know that X[2n+1], X[2n+2], . . . ("the future" if we define the "present" to be time 2n) is conditionally independent of "the past". To find the transition probabilities of the Y-chain, let Q be the transition matrix of the X-chain (labeling the states as 1, 2, . . . , M). Then Q^2 gives the "two-step" transition probabilities of the X-chain. The one-step transitions of the Y-chain correspond to two-step transitions of the X-chain.

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