Comments on the continuity of distribution functions obtained by superposition
HTML articles powered by AMS MathViewer
- by Barthel W. Huff PDF
- Proc. Amer. Math. Soc. 27 (1971), 141-146 Request permission
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.References
- J. R. Blum and Murray Rosenblatt, On the structure of infinitely divisible distributions, Pacific J. Math. 9 (1959), 1–7. MR 105729
- Philip Hartman and Aurel Wintner, On the infinitesimal generators of integral convolutions, Amer. J. Math. 64 (1942), 273–298. MR 6635, DOI 10.2307/2371683
- Barthel W. Huff, The strict subordination of differential processes, Sankhyā Ser. A 31 (1969), 403–412. MR 267633
- Howard G. Tucker, A graduate course in probability, Probability and Mathematical Statistics, Vol. 2, Academic Press, Inc., New York-London, 1967. MR 0221541
- V. M. Zolotarev, Distribution of the superposition of infinitely divisible processes. , Teor. Verojatnost. i Primenen. 3 (1958), 197–200 (Russian, with English summary). MR 0123355
Additional Information
- © Copyright 1971 American Mathematical Society
- 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