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June, 1978 Geometric Ergodicity and R-positivity for General Markov Chains
E. Nummelin, R. L. Tweedie
Ann. Probab. 6(3): 404-420 (June, 1978). DOI: 10.1214/aop/1176995527


We show that for positive recurrent Markov chains on a general state space, a geometric rate of convergence to the stationary distribution $\pi$ in a "small" region ensures the existence of a uniform rate $\rho < 1$ such that for $\pi-\mathrm{a.a.} x, \|P^n(x, \bullet) - \pi(\bullet)\| = O(\rho^n)$. In particular, if there is a point $\alpha$ in the space with $\pi(\alpha) > 0$, the result holds if $|P^n(\alpha, \alpha) - \pi(\alpha)| = O(\rho^n_\alpha)$ for some $\rho_\alpha < 1$. This extends and strengthens the known results on a countable state space. Our results are put in the more general $R$-theoretic context, and the methods we use enable us to establish the existence of limits for sequences $\{R^nP^n(x, A)\}$, as well as exhibiting the solidarity of a geometric rate of convergence for such sequences. We conclude by applying our results to random walk on a half-line.


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E. Nummelin. R. L. Tweedie. "Geometric Ergodicity and R-positivity for General Markov Chains." Ann. Probab. 6 (3) 404 - 420, June, 1978.


Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60051
MathSciNet: MR474504
Digital Object Identifier: 10.1214/aop/1176995527

Primary: 60J10

Keywords: $R$-theory , Ergodic , geometric ergodicity , invariant measure , Markov chain , rate of convergence , splitting

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 3 • June, 1978
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