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Monte Carlo simulation and large deviations theory for uniformly recurrent Markov chains

Published online by Cambridge University Press:  14 July 2016

James A Bucklew ,
Peter Ney  and
John S. Sadowsky
James A Bucklew*
Affiliation:
University of Wisconsin-Madison
Peter Ney*
Affiliation:
University of Wisconsin-Madison
John S. Sadowsky*
Affiliation:
Purdue University
*
∗Postal address: Department of Electrical and Computer Engineering, University of Wisconsin-Madison, WI 53706, USA.
∗∗Department of Mathematics, University of Wisconsin-Madison, WI 53706, USA.
∗∗∗Postal address: School of Electrical Engineering, Purdue University, West Lafayette, IN 47906, USA.
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Abstract

Importance sampling is a Monte Carlo simulation technique in which the simulation distribution is different from the true underlying distribution. In order to obtain an unbiased Monte Carlo estimate of the desired parameter, simulated events are weighted to reflect their true relative frequency. In this paper, we consider the estimation via simulation of certain large deviations probabilities for time-homogeneous Markov chains. We first demonstrate that when the simulation distribution is also a homogeneous Markov chain, the estimator variance will vanish exponentially as the sample size n tends to∞. We then prove that the estimator variance is asymptotically minimized by the same exponentially twisted Markov chain which arises in large deviation theory, and furthermore, this optimization is unique among uniformly recurrent homogeneous Markov chain simulation distributions.

Keywords

IMPORTANCE SAMPLING
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 1 , March 1990 , pp. 44 - 59
Copyright
Copyright © Applied Probability Trust 1990 

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Footnotes

Supported jointly by the Office of Naval Research (N00014-87-K-0155) and the National Science Foundation (ECS-8601071).

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