# Thin tails of fixed points of the nonhomogeneous smoothing transform

Post date: Oct 23, 2015 2:51:13 PM

An article written together with Gerold Alsmeyer available on arXiv.

**Abstract**:

For a given random sequence (C,T1,T2,…) with nonzero C and a.s. finite number of nonzero Tk, the nonhomogeneous 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 the stochastic fixed-point equation (SFPE) 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 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,T1,T2,…) 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.

**Talks given on the subject**:

- Smoothing transform and thin tails, Probability theory and stochastic modeling seminar, Wrocław, Poland, 2015 (slides)
- Exponential moments of fixed points of the nonhomogeneous smoothing transform, Warsaw Summer School in Probability, Warszawa, Poland, 2015 (slides, www)
- Exponential moments of fixed points of the nonhomogeneous smoothing transform, Probability and Analisys, Będlewo, Poland 2015 (slides, www)
- Exponential moments of fixed points of the nonhomogeneous smoothing transform, Functional Analysis seminar, Wrocław, Poland, 2015 (www)