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Importance sampling of heavy-tailed iterated random functions

  • Bohan Chen (a1), Chang-Han Rhee (a1) and Bert Zwart (a1)

Abstract

We consider the stationary solution Z of the Markov chain {Zn}n∈ℕ defined by Zn+1n+1(Zn), where {ψn}n∈ℕ is a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} when x is large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψn such that, for large x, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

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Corresponding author

* Postal address: Stochastics Group, Centrum Wiskunde & Informatica, Science Park 123, 1098 XG, Amsterdam, The Netherlands.
** Email address: b.chen@cwi.nl
*** Current address: Industrial Engineering and Management Sciences, 2145 Sheridan Road, Tech C150, Evanston, Illinois, IL 60208-3109, USA. Email address: chang-han.rhee@northwestern.edu
**** Email address: bert.zwart@cwi.nl

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