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Importance sampling of heavy-tailed iterated random functions

Published online by Cambridge University Press:  16 November 2018

Bohan Chen
Affiliation:
Centrum Wiskunde & Informatica
Chang-Han Rhee
Affiliation:
Centrum Wiskunde & Informatica
Bert Zwart
Affiliation:
Centrum Wiskunde & Informatica

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.

Type
Original Article
Copyright
Copyright © Applied Probability Trust 2018 

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