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Symmetric fixed points of a smoothing transformation
Published online by Cambridge University Press: 22 February 2016
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
- Type
- General Applied Probability
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- Copyright
- Copyright © Applied Probability Trust 2003
Footnotes
Research supported by the German Science Foundation (DFG) grant RO 498/4-1.
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