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An asymptotic sampling distribution for estimators in the previous section can be used to summarize the Monte Carlo variability, provided it is available and the limiting variance is estimable. For IID sampling, moment conditions for the function of interest, , with respect to the target distribution, , suffice. For MCMC sampling, more care needs to be taken to ensure that a limiting distribution holds. We present a subset of the conditions under which the estimators exhibit a normal limiting distribution [9, 13]. The main Markov chain assumption is that of polynomial ergodicity. Let denote the total‐variation distance. Let be the ‐step Markov chain transition kernel, and let such that and for ,

for all . The constant dictates the rate of convergence of the Markov chain. Ergodic Markov chains on finite state spaces are polynomially ergodic. On general state spaces, demonstrating at least polynomial ergodicity usually requires a separate study of the sampler, and we provide some references in Section 6.