Linear stochastic equations in the critical case

Abstract

We consider solutions of the stochastic equation $X\stackrel{d}{=}{\sum }_{i=1}^N{A}_i{X}_i+B$, 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 $m(s)=\mathbb{E}\left[{\sum }_{i=1}^N{A}_i^s\right]$. 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.