Modes of convergence of Markov chain transition probabilities

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Abstract

Suppose {Pn(x, A)} denotes the transition law of a general state space Markov chain {Xn}. We find conditions under which weak convergence of {Xn} to a random variable X with law L (essentially defined by ∝ Pn(x, dy) g(y) → ∝ L(dy) g(y) for bounded continuous g) implies that {Xn} tends to X in total variation (in the sense that ∥ Pn(x,.) - L ∥ → 0), which then shows that L is an invariant measure for {Xn}. The conditions we find involve some irreducibility assumptions on {Xn} and some continuity conditions on the one-step transition law {P(x, A)}. © 1977.

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Tweedie, R. L. (1977). Modes of convergence of Markov chain transition probabilities. Journal of Mathematical Analysis and Applications, 60(1), 280–291. https://doi.org/10.1016/0022-247X(77)90067-1

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