Importance sampling of heavy-tailed iterated random functions

2016/09/11 by Chen, Bohan, Rhee, Chang-Han, Zwart, Bert
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1609.03182

Abstract

We consider a stochastic recurrence equation of the form Z_n+1 = A_n+1 Z_n+B_n+1, where 𝔼[log A₁]<0, 𝔼[log⁺ B₁]x\, when x is large; we provide a consistent simulation estimator using state-dependent importance sampling for the case, where log A₁ is heavy-tailed and the so-called Cramér condition is not satisfied. Our algorithm leads to an estimator for P(Z>x). We show that under natural conditions, our estimator is strongly efficient. Furthermore, we extend our method to the case, where \Z_n\_n∈ℕ is defined via the recursive formula Z_n+1=Ψ_n+1(Z_n) and \Ψ_n\_n∈ℕ is a sequence of i.i.d. random Lipschitz functions.

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