Tail asymptotics of maximums on trees in the critical case

TL;DR

This paper investigates the tail behavior of solutions to a maximum recursion on weighted branching trees in the critical case where the key function is at least one, extending previous power-law results to this boundary scenario.

Contribution

It establishes the asymptotic tail behavior of solutions when the function m(s) is greater than or equal to one, a case not covered by prior work.

Findings

Proves power-law tail asymptotics in the critical case m(s)≥1.

Extends understanding of maximum recursion solutions beyond the subcritical case.

Provides theoretical results for the boundary case in weighted branching trees.

Abstract

We consider solutions to the maximum recursion on weighted branching trees given by$$X\,{\buildrel d\over=}\,\bigvee_{i=1}^{N}{A_iX_i}\vee B,$$where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in\mathbb{N}}$ are random positive numbers and $X_i$ are independent copies of $X$, also independent of $N$, $B$, $\{A_i\}_{i\in\mathbb{N}}$. Properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E}\big[\sum_{i=1}^NA_i^s\big]$. Recently, Jelenkovi\'c and Olvera-Cravioto proved, assuming e.g. $m(s)<1$ for some $s$, that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e.$$\mathbb{P}[R>t]\sim Ct^{-\alpha}$$for some $\alpha>0$ and $C>0$. In this paper we assume $m(s)\ge 1$ for all $s$ and prove analogous results.