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November, 1994 Computable Bounds for Geometric Convergence Rates of Markov Chains
Sean P. Meyn, R. L. Tweedie
Ann. Appl. Probab. 4(4): 981-1011 (November, 1994). DOI: 10.1214/aoap/1177004900

Abstract

Recent results for geometrically ergodic Markov chains show that there exist constants $R < \infty, \rho < 1$ such that $\sup_{|f|\leq V}\big|\int P^n(x, dy)f(y) - \int \pi(dy)f(y)\big| \leq RV(x)\rho^n,$ where $\pi$ is the invariant probability measure and $V$ is any solution of the drift inequalities $\int P(x, dy)V(y) \leq \lambda V(x) + b \mathbb{l}_C(x),$ which are known to guarantee geometric convergence for $\lambda < 1, b < \infty$ and a suitable small set $C$. In this paper we identify for the first time computable bounds on $R$ and $\rho$ in terms of $\lambda, b$ and the minorizing constants which guarantee the smallness of $C$. In the simplest case where $C$ is an atom $\alpha$ with $P(\alpha, \alpha) \geq \delta$ we can choose any $\rho > \vartheta$, where $\lbrack 1 - \vartheta\rbrack^{-1} = \frac{1}{(1 - \lambda)^2} \lbrack 1 - \lambda + b + b^2 + \zeta_\alpha(b(1 - \lambda) + b^2)\rbrack$ and $\zeta_\alpha \leq \big(\frac{32 - 8 \delta^2}{\delta^3}\big) \big(\frac{b}{1 - \lambda}\big)^2,$ and we can then choose $R \leq \rho/(\rho - \vartheta)$. The bounds for general small sets $C$ are similar but more complex. We apply these to simple queuing models and Markov chain Monte Carlo algorithms, although in the latter the bounds are clearly too large for practical application in the case considered.

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Sean P. Meyn. R. L. Tweedie. "Computable Bounds for Geometric Convergence Rates of Markov Chains." Ann. Appl. Probab. 4 (4) 981 - 1011, November, 1994. https://doi.org/10.1214/aoap/1177004900

Information

Published: November, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0812.60059
MathSciNet: MR1304770
Digital Object Identifier: 10.1214/aoap/1177004900

Subjects:
Primary: 60J25

Keywords: Markov chain Monte Carlo , Queueing theory , renewal theory , spectral gap , Uniform convergence

Rights: Copyright © 1994 Institute of Mathematical Statistics

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Vol.4 • No. 4 • November, 1994
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