Precise tail asymptotics of fixed points of the smoothing transform with general weights

Abstract

We consider solutions of the stochastic equation $R=_{d}\sum_{i=1}^{N}A_{i}R_{i}+B$, where $N>1$ is a fixed constant, $A_{i}$ are independent, identically distributed random variables and $R_{i}$ are independent copies of $R$, which are independent both from $A_{i}$’s and $B$. The hypotheses ensuring existence of solutions are well known. Moreover under a number of assumptions the main being $\mathbb{E}|A_{1}|^{\alpha }=1/N$ and $\mathbb{E}|A_{1}|^{\alpha }\log|A_{1}|>0$, the limit $\lim_{t\to\infty }t^{\alpha }\mathbb{P}[|R|>t]=K$ exists. In the present paper, we prove positivity of $K$.