Importance sampling for maxima on trees

Abstract

We study the all-time supremum of the perturbed branching random walk, known to be the endogenous solution to the high-order Lindley equation: W=DmaxY,max1≤i≤N(Wi+Xi),where the {Wi} are independent copies of W, independent of the random vector (Y,N,{Xi}) taking values in R×N×R∞. Under Kesten assumptions, this solution satisfies P(W>t)∼He−αt,t→∞,where α>0 solves the Cramér–Lundberg equation E∑i=1NeαXi=1. This paper establishes the tail asymptotics of W by using the forward iterations of the map defining the fixed-point equation combined with a change of measure along a randomly chosen path. This new approach provides an explicit representation of the constant H and gives rise to unbiased and strongly efficient estimators for the rare event probabilities P(W>t).