On large deviations theory and asymptotically efficient Monte Carlo estimation

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.

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Large Deviations Theory Techniques In Monte Carlo Simulation

1989 Winter Simulation Conference Proceedings ◽

10.1109/wsc.1989.718721 ◽

2005 ◽

Cited By ~ 4

Author(s):

J.S. Sadowsky ◽

J.A. Bucklew

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Large Deviations ◽

Large Deviations Theory

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

Journal of Applied Probability ◽

10.1017/s0021900200038419 ◽

1990 ◽

Vol 27 (01) ◽

pp. 44-59 ◽

Cited By ~ 11

Author(s):

James A Bucklew ◽

Peter Ney ◽

John S. Sadowsky

Keyword(s):

Monte Carlo Simulation ◽

Monte Carlo ◽

Markov Chain ◽

Markov Chains ◽

Large Deviations ◽

Large Deviation ◽

Homogeneous Markov Chain ◽

Underlying Distribution ◽

Monte Carlo Simulation Technique ◽

Large Deviations Theory

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.

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