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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Comments on the continuity of distribution functions obtained by superposition


Author: Barthel W. Huff
Journal: Proc. Amer. Math. Soc. 27 (1971), 141-146
MSC: Primary 60.20
DOI: https://doi.org/10.1090/S0002-9939-1971-0270417-9
MathSciNet review: 0270417
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Abstract: Let $ \{ X(t)\} $ be a differential process and $ Y$ a nonnegative random variable independent of the process. We consider whether the superposition $ X(Y)$ can have a continuous probability distribution. If the process has continuous distributions, then the superposition is continuous if and only if $ P[Y = 0] = 0$. If the process has discontinuous distributions and no trend, then no superposition can have continuous distribution. If the process has discontinuous distributions and nonzero trend, then the superposition onto a random epoch has continuous distribution if and only if $ Y$ has continuous distribution.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0270417-9
Keywords: Superposition, infinitely divisible, Lévy parameters, differential process, random epoch, random sum
Article copyright: © Copyright 1971 American Mathematical Society