On the absolute continuity of fixed points of smoothing transforms
Author:
Sergiy Polots’kiy
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 79 (2009), 139-142
MSC (2000):
Primary 60J80, 60E99; Secondary 60G42
DOI:
https://doi.org/10.1090/S0094-9000-09-00775-3
Published electronically:
December 28, 2009
MathSciNet review:
2494543
Full-text PDF Free Access
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.
References
- J. D. Biggins and D. R. Grey, Continuity of limit random variables in the branching random walk, J. Appl. Probab. 16 (1979), no. 4, 740–749. MR 549554, DOI https://doi.org/10.1017/s0021900200033441
- Aleksander M. Iksanov and Zbigniew J. Jurek, On fixed points of Poisson shot noise transforms, Adv. in Appl. Probab. 34 (2002), no. 4, 798–825. MR 1938943, DOI https://doi.org/10.1239/aap/1037990954
- Quansheng Liu, Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks, Stochastic Process. Appl. 82 (1999), no. 1, 61–87. MR 1695070, DOI https://doi.org/10.1016/S0304-4149%2899%2900008-3
- Quansheng Liu, Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stochastic Process. Appl. 95 (2001), no. 1, 83–107. MR 1847093, DOI https://doi.org/10.1016/S0304-4149%2801%2900092-8
- O. M. Iksanov, Random Series of a Special Form, Branching Random Walks, and Self-Decomposability, Zirka, Kyiv, 2007. (Ukrainian)
References
- J. D. Biggins and D. R. Grey, Continuity of limit random variables in the branching random walk, J. Appl. Prob. 16 (1979), 740–749. MR 549554 (80j:60107)
- A. Iksanov and Z. Jurek, On fixed points of Poisson shot noise transforms, Adv. Appl. Prob. 34 (2002), 798–825. MR 1938943 (2003i:60021)
- Q. Liu, Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random walks, Stoch. Proc. Appl. 82 (1999), 61–87. MR 1695070 (2001b:60103)
- Q. Liu, Asymptotic properties and absolute continuity of laws stable by random weighted mean, Stoch. Proc. Appl. 95 (2001), 83–107. MR 1847093 (2002e:60141)
- O. M. Iksanov, Random Series of a Special Form, Branching Random Walks, and Self-Decomposability, Zirka, Kyiv, 2007. (Ukrainian)
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2000):
60J80,
60E99,
60G42
Retrieve articles in all journals
with MSC (2000):
60J80,
60E99,
60G42
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
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
