Tail Asymptotics for Maximal SFPEs
Abstract: Different types of branching, measure-valued recursions known as stochastic fixed-point equations (SFPEs) show up in various applications, often characterizing some limiting distribution. We look at a max-type SFPE, which is nice because an important solution, the random variable W, is the all-time maximum of a branching random walk with a perturbation. It has been known that under Cramer-Lundberg conditions the right tail of W is asymptotically exponential, something initially proved by generalizing the implicit renewal theory of Goldie (1991). This leads to the asymptotics characterized by a constant that is “implicit,” in the sense that it is a functional of W itself. The tail asymptotics of (non-branching) random walks were historically established using change-of-measure and direct renewal-theoretic arguments, and we extend this framework to the branching case, obtaining an alternate representation for the asymptotic constant in terms of the underlying branching process. The change of measure used is also of independent interest, and allows for an unbiased, strongly efficient accelerated Monte Carlo estimator for tail probabilities of W, much in the spirit of Siegmund (1976). My goal is to make this talk accessible to graduate students with a background or interest in probability.