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Numerical integration using V-uniformly ergodic Markov chains

Part of: Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  14 July 2016

Peter Mathé
Peter Mathé*
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics
*
∗ Postal address: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany. Email address: [email protected]
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Abstract

We study numerical integration based on Markov chains. Our focus is on establishing error bounds uniformly on classes of integrands. Since in general state space the concept of uniform ergodicity is too restrictive to cover important cases, we analyze the error of V-uniformly ergodic Markov chains. We place emphasis on the interplay between ergodicity properties of the transition kernel, the initial distributions and the classes of integrands. Our analysis is based on arguments from interpolation theory.

Keywords

Numerical integrationV-uniformly ergodic Markov chaininterpolation

MSC classification

Primary: 65C05: Monte Carlo methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 41 , Issue 4 , December 2004 , pp. 1104 - 1112
Copyright
Copyright © Applied Probability Trust 2004 

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References

Banach, S. (1987). Theory of Linear Operations (North-Holland Math. Library 38). North-Holland, Amsterdam.Google Scholar
Bergh, J. and Löfström, J. (1976). Interpolation Spaces. An Introduction (Grundlehren der Mathematischen Wissenschaften 223). Springer, Berlin.Google Scholar
Jarner, S. F., and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stoch. Process. Appl. 85, 341–361.Google Scholar
Krengel, U. (1985). Ergodic Theorems (De Gruyter Stud. in Math. 6). De Gruyter, Berlin.Google Scholar
Mathé, P. (1999). Numerical integration using Markov chains. Monte Carlo Meth. Appl. 5, 325–343.CrossRefGoogle Scholar
Meyn, S. P., and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Revuz, D. (1975). Markov Chains (North-Holland Math. Library 11). North-Holland, Amsterdam.Google Scholar
Tierney, L. (1998). A note on Metropolis-Hastings kernels for general state spaces. Ann. Appl. Prob. 8, 1–9.CrossRefGoogle Scholar