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Central Limit Theorem for Adaptive Multilevel Splitting Estimators in an Idealized Setting

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS,volume 163)

Abstract

The Adaptive Multilevel Splitting (AMS) algorithm is a powerful and versatile iterative method to estimate the probabilities of rare events. We prove a new central limit theorem for the associated AMS estimators introduced in [5], and which have been recently revisited in [3]—the main result there being (non-asymptotic) unbiasedness of the estimators. To prove asymptotic normality, we rely on and extend the technique presented in [3]: the (asymptotic) analysis of an integral equation. Numerical simulations illustrate the convergence and the construction of Gaussian confidence intervals.

Keywords

  • Monte-Carlo simulation
  • Rare events
  • Multilevel splitting
  • Central limit theorem

Mathematics Subject Classification:

  • 65C05
  • 65C35
  • 60F05

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Acknowledgments

C.-E. B. would like to thank G. Samaey, T. Lelièvre and M. Rousset for the invitation to give a talk on the topic of this paper at the 11th MCQMC Conference, in the special session on Mathematical aspects of Monte Carlo methods for molecular dynamics. We would also like to thank the referees for suggestions which improved the presentation of the paper.

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Correspondence to Charles-Edouard Bréhier .

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Bréhier, CE., Goudenège, L., Tudela, L. (2016). Central Limit Theorem for Adaptive Multilevel Splitting Estimators in an Idealized Setting. In: Cools, R., Nuyens, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods. Springer Proceedings in Mathematics & Statistics, vol 163. Springer, Cham. https://doi.org/10.1007/978-3-319-33507-0_10

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