2014/02/17 by Alexander Iksanov, Matthias Meiners, Iksanov, Alexander +1 · 2 citations
Decision Sciences · Mathematics · #39B32 #60J80 #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.1402.4147
openalex publication_date 2014/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a sequence (C1,…,Cd,T1,T2,…) of real-valued random variables with N := #\j ≥ 1: Tj \not = 0\ < ∞ almost surely, there is an associated smoothing transformation which maps a distribution P on ℝd to the distribution of ∑j ≥ 1 Tj X(j) + C where C = (C1,…,Cd) and (X(j))j ≥ 1 is a sequence of independent random vectors with distribution P independent of (C1,…,Cd,T1,T2,…). We are interested in the fixed points of this mapping. By improving on the techniques developed in [G. Alsmeyer, J.D. Biggins, and M. Meiners. The functional equation of the smoothing transform \em Ann. Probab., 40(5):2069--2105, 2012] and [G. Alsmeyer and M. Meiners. Fixed points of the smoothing transform: two-sided solutions. \em Probab. Theory Related Fields, 155(1-2):165--199, 2013], we determine the set of all fixed points under weak assumptions on (C1,…,Cd,T1,T2,…). In contrast to earlier studies, this includes the most intricate case when the Tj take both positive and negative values with positive probability. In this case, in some situations, the set of fixed points is a subset of the corresponding set when the Tj are replaced by their absolute values, while in other situations, additional solutions arise.