On the tail distribution of the solution to some law equation

TL;DR

This paper investigates the tail distribution of solutions to a specific distribution equation related to branching processes, providing a new method to analyze tail behavior based on joint tail analysis of involved variables.

Contribution

It introduces a novel direct approach to determine the tail distribution of solutions to a generalized distribution equation, expanding on prior work by Bertoin.

Findings

Derived the tail behavior of the solution for a generalized distribution equation.

Provided a new method based on joint tail analysis of the variables.

Extended understanding of distribution equations related to branching processes.

Abstract

We consider a distribution equation which was initially studied by Bertoin \cite{Bertoin}: \[M \stackrel{d}{=} \max\{\widetilde{\nu}, \max_{1\leq k\leq \nu}M_k\}.\] where $\{M_k\}_{k\geq 1}$ are i.i.d. copies of $M$ and independent of $(\widetilde{\nu}, \nu)\in\mathbb{R}_+\times\mathbb{N}$. We obtain the tail behaviour of the solution of a generalised equation in a different but direct method by considering the joint tail of $(\widetilde{\nu}, \nu)$.