Importance sampling of heavy-tailed iterated random functions. (16th November 2018)

• Record Type: Journal Article
• Authors: Chen, Bohan; Rhee, Chang-Han; Zwart, Bert
• Is Part Of: Advances in applied probability. Volume 50:Number 3(2018)
• Page Start: 805
• Page End: 832
• Publication Date: 2018-11-16
• Subjects: State-dependent importance sampling, -- heavy-tailed distribution, -- iterated random function, -- perpetuities; Primary 65C05, -- Secondary 60J05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2018.37
• Languages: English
• ISSNs: 0001-8678

Abstract

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 {ψ n } 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.