Fixed points of the smoothing transform: Two-sided solutions

2010/09/13 by Alsmeyer, Gerold, Meiners, Matthias
#60E05 (Primary) 39B32 #60E10 #60J80 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1009.2412

Abstract

Given a sequence (C,T) = (C,T₁,T₂,...) 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 P(\R) of probability distributions on the real line, maps an element P\inP(\R) to the law of C + ∑_{j ≥ 1} T_j X_j, 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 X₁\stackreld=C + ∑_{j ≥ 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 \citeFJ2000 and Aldous and Bandyopadhyay \citeAB2005. 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.

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