publication . Article . Other literature type . Preprint . 2008

Efficient rare-event simulation for the maximum of heavy-tailed random walks

Blanchet, Jose; Glynn, Peter;
Open Access
  • Published: 01 Aug 2008 Journal: The Annals of Applied Probability, volume 18, pages 1,351-1,378 (issn: 1050-5164, Copyright policy)
  • Publisher: Institute of Mathematical Statistics
Abstract
Let $(X_n:n\geq 0)$ be a sequence of i.i.d. r.v.'s with negative mean. Set $S_0=0$ and define $S_n=X_1+... +X_n$. We propose an importance sampling algorithm to estimate the tail of $M=\max \{S_n:n\geq 0\}$ that is strongly efficient for both light and heavy-tailed increment distributions. Moreover, in the case of heavy-tailed increments and under additional technical assumptions, our estimator can be shown to have asymptotically vanishing relative variance in the sense that its coefficient of variation vanishes as the tail parameter increases. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Si...
Subjects
free text keywords: Statistics, Probability and Uncertainty, Statistics and Probability, Mathematical optimization, Independent and identically distributed random variables, Relative standard deviation, Estimator, Coefficient of variation, Importance sampling, Rare event simulation, Random variable, Combinatorics, Random walk, Mathematics, State-dependent importance sampling, rare-event simulation, heavy-tails, Lyapunov bounds, random walks, single-server queue, change-of-measure, 60G50, 60J05, 68W40, 60G70, 60J20, Mathematics - Probability, 60G50, 60J05, 68W40 (Primary) 60G70, 60J20 (Secondary)

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Department of Management Science and Engineering Stanford University Stanford, California 94305 USA

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