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I have encountered one last problem with regarding to the Metropolis-Hastings algorithm. I know that ergodicity is needed in the algorithm to imply convergence to a unique stationary distribution. But how is ergodicity proved in the M-H algorithm? Thank you for your help

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You construct a chain $P$ that is derivated from a chain $Q$ with an accept/reject condition.

When $Q$ is irreductible and positively recurrent, so is $P$; now as $P(x,x)>0$ for at least one $X$, $P$ is aperiodic, so $P$ is ergodic.

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