Hostname: page-component-6d856f89d9-4thr5 Total loading time: 0 Render date: 2024-07-16T07:00:18.170Z Has data issue: false hasContentIssue false
  • English
  • Français

Rare-Event Simulation of Heavy-Tailed Random Walks by Sequential Importance Sampling and Resampling

Part of: Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  04 January 2016

Hock Peng Chan ,
Shaojie Deng  and
Tze-Leung Lai
Hock Peng Chan*
Affiliation:
National University of Singapore
Shaojie Deng
Affiliation:
Microsoft
Tze-Leung Lai*
Affiliation:
Stanford University
*
∗ Postal address: Department of Statistics and Applied Probability, National University of Singapore, Science Drive 1, 119260, Singapore.
∗∗ Postal address: Statistics Department, Stanford University, Stanford, CA 94305, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a new approach to simulating rare events for Markov random walks with heavy-tailed increments. This approach involves sequential importance sampling and resampling, and uses a martingale representation of the corresponding estimate of the rare-event probability to show that it is unbiased and to bound its variance. By choosing the importance measures and resampling weights suitably, it is shown how this approach can yield asymptotically efficient Monte Carlo estimates.

Keywords

Efficient simulationheavy-tailed distributionsequential Monte Carloregularly varying tail

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 60G50: Sums of independent random variables; random walks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 1173 - 1196
Copyright
© Applied Probability Trust 

References

Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Kroese, D. P. (2006). Improved algorithms for rare event simulation with heavy tails. Adv. Appl. Prob. 38, 545–558.Google Scholar
Asmussen, S., Binswanger, K. and Hojgaard, B. (2000). Rare events simulation for heavy-tailed distributions. Bernoulli 6, 303–322.Google Scholar
Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378.CrossRefGoogle Scholar
Blanchet, J. and Lam, H. (2012). State-dependent importance sampling for rare-event simulation: an overview and recent advances. Surveys Operat. Res. Manag. Sci. 17, 38–59.Google Scholar
Blanchet, J. and Liu, J. (2006). Efficient simulation for large deviation probabilities of sums of heavy-tailed increments. In Proc. 2006 Winter Simul. Conf., IEEE, pp. 757–764.Google Scholar
Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Adv. Appl. Prob. 40, 1104–1128.CrossRefGoogle Scholar
Blanchet, J., Juneja, S. and Rojas-Nandayapa, L. (2008). Efficient tail estimation for sums of correlated lognormals. In Proc. 2008 Winter Simul. Conf., IEEE, pp. 607–614.CrossRefGoogle Scholar
Chan, H. P. and Lai, T. L. (2011). A sequential Monte Carlo approach to computing tail probabilities in stochastic models. Ann. Appl. Prob. 21, 2315–2342.Google Scholar
Chow, Y. S. and Lai, T. L. (1975). Some one-sided theorems on the tail distribution of sample sums with applications to the last time and largest excess of boundary crossings. Trans. Amer. Math. Soc. 208, 51–72.CrossRefGoogle Scholar
Chow, Y. S. and Lai, T. L. (1978). Paley-type inequalities and convergence rates related to the law of large numbers and extended renewal theory. Z. Wahrscheinlichkeitsth. 45, 1–19.CrossRefGoogle Scholar
Chow, Y. S. and Teicher, H. (1988). Probability Theory: Independence, Interchangeability, Martingales, 2nd edn. Springer, New York.CrossRefGoogle Scholar
Del Moral, P. and Garnier, J. (2005). Genealogical particle analysis of rare events. Ann. Appl. Prob. 15, 2496–2534.Google Scholar
Dupuis, P., Leder, K. and Wang, H. (2007). Importance sampling for sums of random variables with regularly varying tails. ACM Trans. Model. Comput. Simul. 17, 21pp.Google Scholar
Foss, S., Korshunov, D. and Zachary, S. (2009). An Introduction to Heavy-tailed and Subexponential Distributions. Springer, New York.Google Scholar
Juneja, S. (2007). Estimating tail probabilities of heavy tailed distributions with asymptotically zero relative error. Queueing Systems 57, 115–127.Google Scholar
Kong, A., Liu, J. S. and Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89, 278–288.Google Scholar
Rozovskiǐ, L. V. (1989). Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law. Theory Prob. Appl. 34, 625–644.Google Scholar