Importance sampling of heavy-tailed iterated random functions. (16th November 2018) - Chen, Bohan; Rhee, Chang-Han; Zwart, Bert
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.