Importance sampling of heavy-tailed iterated random functions

Bohan Chen; Chang Han Rhee; Bert Zwart

doi:10.1017/apr.2018.37

Importance sampling of heavy-tailed iterated random functions

Bohan Chen, Chang Han Rhee, Bert Zwart

Stochastic Operations Research

Research output: Contribution to journal › Article › Academic › peer-review

2 Citations (Scopus)

Abstract

We consider the stationary solution Z of the Markov chain {Z_n}n∈N defined by Z_n+1 = Ψ_n+1(Z_n), 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 Ψ_n 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.

Original language	English
Pages (from-to)	805-832
Number of pages	28
Journal	Advances in Applied Probability
Volume	50
Issue number	3
DOIs	https://doi.org/10.1017/apr.2018.37
Publication status	Published - 1 Sept 2018

Bibliographical note

Funding Information:
The authors gratefully acknowledge the support from the Netherlands Organization for Scientific Research (NWO) through the Vici grant 639.033.413.

Publisher Copyright:
Copyright © Applied Probability Trust 2018.

Funding

The authors gratefully acknowledge the support from the Netherlands Organization for Scientific Research (NWO) through the Vici grant 639.033.413.

Keywords

heavy-tailed distribution
iterated random function
perpetuities
State-dependent importance sampling

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10.1017/apr.2018.37

Cite this

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title = "Importance sampling of heavy-tailed iterated random functions",

abstract = "We consider the stationary solution Z of the Markov chain \{Zn\}n∈N defined by Zn+1 = Ψn+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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N2 - We consider the stationary solution Z of the Markov chain {Zn}n∈N defined by Zn+1 = Ψn+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.

AB - We consider the stationary solution Z of the Markov chain {Zn}n∈N defined by Zn+1 = Ψn+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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KW - iterated random function

KW - perpetuities

KW - State-dependent importance sampling

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JO - Advances in Applied Probability

JF - Advances in Applied Probability

IS - 3

ER -

Importance sampling of heavy-tailed iterated random functions

Abstract

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Keywords

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