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Fixed points of the smoothing transform: Two-sided solutions
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
2013 (English)In: Probability theory and related fields, ISSN 0178-8051, E-ISSN 1432-2064, Vol. 155, no 1-2, 165-199 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2013. Vol. 155, no 1-2, 165-199 p.
Keyword [en]
Branching random walk, Characteristic function, General branching processes, Infinite divisibility, Multiplicative martingales, Smoothing transformation, Stable distribution, Stochastic fixed-point equation, Weighted branching process
National Category
URN: urn:nbn:se:uu:diva-194913DOI: 10.1007/s00440-011-0395-yISI: 000313736500005OAI: oai:DiVA.org:uu-194913DiVA: diva2:606649
Available from: 2013-02-20 Created: 2013-02-19 Last updated: 2013-02-25Bibliographically approved

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