On the renewal theorem for maxima on trees

Speaker: Zbigniew Palmowski 
Affiliation: Wroclaw University of Science and Technology

Abstract

We consider the high-order Lindley equation with a solution W. It is known that under Kesten assumptions, the distribution of W has the tail which is asymptotically exponential. The main goal of this talk is to provide an explicit representation for P(W > t), which can be directly connected to the underlying weighted branching process where W is constructed and that can be used to construct unbiased and strongly efficient estimators for all t. Furthermore, we show how this new representation can be directly analyzed using Alsmeyer's Markov renewal theorem, yielding an alternative representation for the constant appearing before the exponential term. We provide numerical examples illustrating the use of this new algorithm. The talk is based on a joint paper with B. Basrak, M. Conroy and M. Olvera-Cravioto.