@article{ALSMEYER20173014,
title = {Thin tails of fixed points of the nonhomogeneous smoothing transform},
journal = {Stochastic Processes and their Applications},
volume = {127},
number = {9},
pages = {3014-3041},
year = {2017},
issn = {0304-4149},
doi = {https://doi.org/10.1016/j.spa.2017.01.008},
url = {https://www.sciencedirect.com/science/article/pii/S0304414917300236},
author = {Gerold Alsmeyer and Piotr Dyszewski},
keywords = {Nonhomogeneous smoothing transform, Stochastic fixed point, Moment generating function, Exponential moment, Poissonian tail, Weighted branching process, Forward and backward equation, Quicksort distribution},
abstract = {For a given random sequence (C,T1,T2,…), the smoothing transform S maps the law of a real random variable X to the law of ∑k≥1TkXk+C, where X1,X2,… are independent copies of X and also independent of (C,T1,T2,…). This law is a fixed point of S if X=d∑k≥1TkXk+C holds true, where =d denotes equality in law. Under suitable conditions including EC=0, S possesses a unique fixed point within the class of centered distributions, called the canonical solution because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on (C,T1,T2,…) such that the canonical solution exhibits right and/or left Poissonian 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.}
}