# PADS Seminar: Renewal Theory for Maxima on Trees

**Author: Lona**

**Author: Lona**

**Speaker: Michael Conroy (University of Arizona)**

**Abstract: **This talk focuses on the all-time supremum W of the perturbed branching random walk, which is the so-called endogenous solution to the stochastic fixed-point equation known as the high-order Lindley equation. Under certain assumptions (in particular that the paths of each branch of the random walk have negative drift), W satisfies the tail asymptotic P(W > t) ~ He^{-at} for certain constants H, a > 0. Using tools from Markov renewal theory and spine changes of measure for branching processes, we establish the tail asymptotic by analyzing the forward iterations of the map defining the fixed-point equation. As a bonus to this approach, we obtain an unbiased, strongly-efficient, and easy-to-simulate estimator for the rare event probability P(W > t). This talk presents joint work with Bojan Basrak (University of Zagreb), Mariana Olvera-Cravioto (UNC Chapel Hill), and Zbigniew Palmowski (Wroclaw University of Science and Technology).