# A Markov chain is said to be aperiodic9 if and only if there exists a specific time t > 1 such…

A Markov chain is said to be aperiodic9 if and only if there exists a specific time t > 1 such that for every pair of states x and x’, we have P(Xt =x’ |X1 =x) > 0. If a Markov chain is aperiodic, it will also be irreducible and, hence, have a unique stationary distribution, say Pr(X). 10 Moreover, if the chain is aperiodic, it will converge to its stationary distribution. That is,

for all states x and x’. Hence, when simulating the Markov chain, the simulated instantiations are eventually sampled from the stationary distribution Pr(X) and become independent of the initial state at time 1.

Consider now the Markov chain for a binary variable X with the transition matrix P(x|x¯) = 1 and P(x¯|x) = 1; hence, P(x¯|x¯) = 0 and P(x|x) = 0. Is this chain aperiodic? Is it irreducible? If it is, identify its unique stationary distribution. Will the chain converge to any distribution?

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