Maximums on Trees

TL;DR

This paper analyzes the tail behavior of the minimal endogenous solution to a maximum recursion on weighted branching trees, revealing a power-law decay under natural conditions, with implications for related branching processes.

Contribution

It establishes the power-law asymptotics for the solution to a maximum recursion on weighted branching trees, extending understanding of tail behaviors in branching recursions.

Findings

The tail probability of the solution decays as a power-law.

The asymptotic behavior is characterized by specific parameters nd H.

Implications for the tail behavior of related branching recursions.

Abstract

We study the minimal/endogenous solution $R$ to the maximum recursion on weighted branching trees given by $$R\stackrel{\mathcal{D}}{=}\left(\bigvee_{i=1}^NC_iR_i \right)\vee Q,$$ where $(Q,N,C_1,C_2,\dots)$ is a random vector with $N\in \mathbb{N}\cup\{\infty\}$, $P(|Q|>0)>0$ and nonnegative weights $\{C_i\}$, and $\{R_i\}_{i\in\mathbb{N}}$ is a sequence of i.i.d. copies of $R$ independent of $(Q,N,C_1,C_2,\dots)$; $\stackrel{\mathcal{D}}{=}$ denotes equality in distribution. Furthermore, when $Q>0$ this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of $R$ is power-law, i.e., $P(|R|>x)\sim Hx^{-\alpha}$, for some $\alpha>0$ and $H>0$. This has direct implications for the tail behavior of other well known branching recursions.