Thin tails of fixed points of the nonhomogeneous smoothing transform

Abstract

For a given random sequence $(C,T_1,T_2,\dots )$ 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 $X_1,X_2,\dots$ are independent copies of $X$ and also independent of $(C,T_1,T_2,\dots )$. 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 $(C,T_1,T_2,\dots )$ 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.

26 pages

References in corpus (1)

• A survey of max-type recursive distributional equations