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I am aware this question borderlines retardedness, but I am seeking an accurate explanation. I understand in null-recurrent cases, the expected amount of time to explore states can be infinite. Is this because if the expected time that you visit states is infinite, it will "take infinite amount of time, on average, for the Markov Chain to converge"? Is there anyway of formalising this using the total variation distance of measures?

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The goal of MCMC techniques is to compute integrals with respect to a probability measure $\pi$ using some Markov process with stationary distribution $\pi$. But null recurrent Markov processes have no stationary distribution. Hence MCMC techniques are, by their very construction, restricted to positive recurrent Markov processes.

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