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Fixed points of the smoothing transform: two-sided solutions
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  • Published: 21 October 2011

Fixed points of the smoothing transform: two-sided solutions

  • Gerold Alsmeyer1 &
  • Matthias Meiners2 

Probability Theory and Related Fields volume 155, pages 165–199 (2013)Cite this article

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Abstract

Given a sequence (C, T) = (C, T 1, T 2, . . .) of real-valued random variables with T j ≥ 0 for all j ≥ 1 and almost surely finite N = sup{j ≥ 1 : T j > 0}, the smoothing transform associated with (C, T), defined on the set \({\mathcal{P}(\mathbb R)}\) of probability distributions on the real line, maps an element \({P \in \mathcal{P}(\mathbb R)}\) to the law of \({C + \sum_{j \geq 1} T_j X_j}\) , where X 1, X 2, . . . 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 \({X_{1} \stackrel {\mathrm{d}}{=}C + \sum_{j \geq 1} T_j X_j}\) . 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 \({\tt Quicksort}\) equation.

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Author information

Authors and Affiliations

  1. Institut für Mathematische Statistik, Universität Münster, Einsteinstraße 62, 48149, Münster, Germany

    Gerold Alsmeyer

  2. Matematiska institutionen, Uppsala universitet, Box 480, 751 06, Uppsala, Sweden

    Matthias Meiners

Authors
  1. Gerold Alsmeyer
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  2. Matthias Meiners
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Corresponding author

Correspondence to Matthias Meiners.

Additional information

M. Meiners’ research was supported by DFG-grant Me 3625/1-1.

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Alsmeyer, G., Meiners, M. Fixed points of the smoothing transform: two-sided solutions. Probab. Theory Relat. Fields 155, 165–199 (2013). https://doi.org/10.1007/s00440-011-0395-y

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  • Received: 13 September 2010

  • Revised: 11 October 2011

  • Published: 21 October 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0395-y

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Keywords

  • Branching random walk
  • Characteristic function
  • General branching processes
  • Infinite divisibility
  • Multiplicative martingales
  • Smoothing transformation
  • Stable distribution
  • Stochastic fixed-point equation
  • Weighted branching process

Mathematics Subject Classification (2010)

  • Primary 60E05
  • Secondary 39B32
  • 60E10
  • 60J80
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