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

TL;DR

This paper investigates the tail behavior of solutions to a stochastic fixed point equation, establishing the positivity of the tail constant under general conditions, which is crucial for understanding the distribution's tail decay.

Contribution

It proves the positivity of the tail constant for solutions of the smoothing transform with general weights, extending previous results to broader conditions.

Findings

The tail constant K is positive under specified conditions.

The tail asymptotics follow a power-law decay with exponent alpha.

The results apply to a wide class of stochastic fixed point equations.

Abstract

We consider solutions of the stochastic equation $R=_d\sum_{i=1}^NA_iR_i+B$, 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 $K$.