Linear stochastic equations in the critical case

TL;DR

This paper investigates the existence and asymptotic behavior of solutions to a critical stochastic linear equation where the key function is tangent to one, revealing new insights in this borderline case.

Contribution

It provides the first detailed analysis of solutions in the critical case where the function m(s) is tangent to y=1, including existence results and asymptotic descriptions.

Findings

Existence of solutions under critical conditions

Asymptotic behavior characterized in the critical case

Conditions for solutions when m(s) is tangent to y=1

Abstract

We consider solutions of the stochastic equation $X \stackrel{d}= \sum_{i=1}^N A_iX_i + B$, where $N$ is a random natural number, $B$ and $A_i$ are random positive numbers and $X_i$ are independent copies of $X$, which are independent also of $N,B,A_i$. Properties of solutions of this equation are mainly coded in the function $m(s)=\mathbb{E}\big[\sum_{i=1}^N A_i^s \big]$. In this paper we study the critical case when the function $m$ is tangent to the line $y=1$. Then, under a number of further assumptions, we prove existence of solutions and describe their asymptotic behavior.