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Symmetric fixed points of a smoothing transformation

Part of: Markov processes Distribution theory - Probability Limit theorems

Published online by Cambridge University Press:  22 February 2016

Amke Caliebe
Amke Caliebe*
Affiliation:
Christian-Albrechts-Universität zu Kiel
*
∗ Postal address: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany. Email address: caliebe@math.uni-kiel.de
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Abstract

Let T = (T1, T2,…) be a sequence of real random variables with ∑j=1∞1|Tj|>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑j=1∞TjWj, where W, W1, W2,… have distribution μ and T, W1, W2,… are independent. We show that the representation of general solutions is a mixture of certain infinitely divisible distributions. This result can be applied to investigate the existence of symmetric solutions for Tj ≥ 0: essentially under the condition that E ∑j=1∞Tj2 log+Tj2 < ∞, the existence of nontrivial symmetric solutions is exactly determined, revealing a connection with the existence of positive solutions of a related fixed-point equation. Furthermore, we derive results about a special class of canonical symmetric solutions including statements about Lebesgue density and moments.

Keywords

Distributional fixed-point equationfunctional equationconvergence of triangular schemesrepresentation of solutionsinfinitely divisible distributionstable distribution

MSC classification

Primary: 60E10: Characteristic functions; other transforms 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 377 - 394
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

Research supported by the German Science Foundation (DFG) grant RO 498/4-1.

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