Thin tails of fixed points of the nonhomogeneous smoothing transform

TL;DR

This paper investigates the tail behavior of fixed points of the nonhomogeneous smoothing transform, providing conditions for Poisson tails and analyzing the moment generating function's convergence, with applications to the Quicksort distribution.

Contribution

It establishes new conditions under which the fixed points have Poisson tails and determines the moment generating function's abscissa of convergence.

Findings

Fixed points can have right and/or left Poisson tails.

The abscissa of convergence of the moment generating function is characterized.

Application to the tail behavior of the Quicksort distribution.

Abstract

For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\sum_{k\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\ldots$ are independent copies of $X$ and also independent of $(C,T_{1},T_{2},\ldots)$. This law is a fixed point of $\mathcal{S}$ if the stochastic fixed-point equation (SFPE) $X\stackrel{d}{=}\sum_{k\ge 1}T_{k}X_{k}+C$ holds true, where $\stackrel{d}{=}$ denotes equality in law. Under suitable conditions including $\mathbb{E} C=0$, $\mathcal{S}$ possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on $(C,T_{1},T_{2},\ldots)$ such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found.