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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

On the absolute continuity of fixed points of smoothing transforms


Author: Sergiy Polots'kiy
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 79 (2008).
Journal: Theor. Probability and Math. Statist. 79 (2009), 139-142
MSC (2000): Primary 60J80, 60E99; Secondary 60G42
Published electronically: December 28, 2009
MathSciNet review: 2494543
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Abstract | References | Similar Articles | Additional Information

Abstract: Let the distribution of a nonnegative random variable $ W$ be such that $ W \overset{d} = \sum_{i=1}^J Y_i W_i$, where $ \{Y_i\colon i=1,\dots,J\}$ are some positive random variables. Under some moment conditions imposed on $ Y_i$ we show that the distribution of $ W$ is a mixture of the atom at the origin and an absolutely continuous component.


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Additional Information

Sergiy Polots'kiy
Affiliation: Faculty for Cybernetics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine
Email: pilot_ser@mail.ru

DOI: http://dx.doi.org/10.1090/S0094-9000-09-00775-3
PII: S 0094-9000(09)00775-3
Keywords: Smoothing transform, absolute continuity, fixed points
Received by editor(s): February 27, 2008
Published electronically: December 28, 2009
Article copyright: © Copyright 2009 American Mathematical Society