Fixed points of the smoothing transform: Two-sided solutions

Abstract

Given a sequence (C, T) = (C, T1, T2, . . .) of real-valued random variables with Tj ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : Tj > 0}, the smoothing transform associated with (C, T), defined on the set P(ℝ) of probability distributions on the real line, maps an element P ∈ P(ℝ) to the law of, where X1, X2, . . . is a sequence of i. i. d. random variables independent of (C, T) and with distribution P. We study the fixed points of the smoothing transform, that is, the solutions to the stochastic fixed-point equation. By drawing on recent work by the authors with J. D. Biggins, a full description of the set of solutions is provided under weak assumptions on the sequence (C, T). This solves problems posed by Fill and Janson (Electron Commun Probab 5:77-84, 2000) and Aldous and Bandyopadhyay (Ann Appl Probab 15(2):1047-1110, 2005). Our results include precise characterizations of the sets of solutions to large classes of stochastic fixed-point equations that appear in the asymptotic analysis of divide-and-conquer algorithms, for instance the Quicksort equation. © 2011 Springer-Verlag.