We consider the stationary solution Z of the Markov chain {Zn}nϵℕ defined by Zn+1n+1(Zn), where {ψn}nϵ 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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Advances in Applied Probability
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Chen, B., Rhee, C.-H., & Zwart, B. (2018). Importance sampling of heavy-tailed iterated random functions. Advances in Applied Probability, 50(3), 805–832. doi:10.1017/apr.2018.37