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Importance sampling for maxima on trees

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  • Basrak, Bojan
  • Conroy, Michael
  • Olvera-Cravioto, Mariana
  • Palmowski, Zbigniew

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).

Suggested Citation

  • Basrak, Bojan & Conroy, Michael & Olvera-Cravioto, Mariana & Palmowski, Zbigniew, 2022. "Importance sampling for maxima on trees," Stochastic Processes and their Applications, Elsevier, vol. 148(C), pages 139-179.
    • Handle: RePEc:eee:spapps:v:148:y:2022:i:c:p:139-179
    DOI: 10.1016/j.spa.2022.02.005
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    References listed on IDEAS

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