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The Smoothing Transform: A Review of Contraction Results

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Random Matrices and Iterated Random Functions

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 53))

Abstract

Given a sequence (C, T) = (C, T 1, T 2, …) of real-valued random variables, the associated so-called smoothing transform \(\mathcal{S}\) maps a distribution F from a subset Γ of distributions on \(\mathbb{R}\) to the distribution of \(\sum _{i\geq 1}T_{i}X_{i} + C\), where X 1, X 2, … are iid with common distribution F and independent of (C, T). This review aims at providing a comprehensive account of contraction properties of \(\mathcal{S}\) on subsets Γ specified by the existence of moments up to a given order like, for instance, \({\mathcal{P}}^{p}(\mathbb{R}) =\{ F:\int \vert x{\vert }^{p}\,F(dx) < \infty \}\) for p > 0 or \(\mathcal{P}_{c}^{p}(\mathbb{R}) =\{ F \in {\mathcal{P}}^{p}(\mathbb{R}):\int x\,F(dx) = c\}\) for p ≥ 1. The metrics used here are the minimal ℓ p -metric and the Zolotarev metric ζ p , both briefly introduced in Sect. 3.

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Acknowledgements

G.A. was supported by Deutsche Forschungsgemeinschaft (SFB 878).

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

    Gerold Alsmeyer

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  1. Gerold Alsmeyer
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  1. Westphalian Wilhelms University Münster Institute of Mathematics and Informatics, Münster, Germany

    Gerold Alsmeyer

  2. Westphalian Wilhelms University Münster Institute for Mathematical Statistics, Münster, Germany

    Matthias Löwe

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Alsmeyer, G. (2013). The Smoothing Transform: A Review of Contraction Results. In: Alsmeyer, G., Löwe, M. (eds) Random Matrices and Iterated Random Functions. Springer Proceedings in Mathematics & Statistics, vol 53. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38806-4_9

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