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Symmetric fixed points of a smoothing transformation

Published online by Cambridge University Press:  22 February 2016

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: [email protected]

Abstract

Let T = (T1, T2,…) be a sequence of real random variables with ∑j=11|Tj|>0 < ∞ almost surely. We consider the following equation for distributions μ: W ≅ ∑j=1TjWj, 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=1Tj2 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.

Type
General Applied Probability
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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