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Characterization of stable processes by identically distributed stochastic integrals

Published online by Cambridge University Press:  01 July 2016

M. Riedel
M. Riedel*
Affiliation:
Karl-Marx-Universität Leipzig
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Abstract

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

Keywords

WIENER PROCESSINFINITELY DIVISIBLE DISTRIBUTIONLÉVY CANONICAL REPRESENTATIONCONTINUOUSHOMOGENEOUS PROCESS WITH INDEPENDENT INCREMENTSPHRAGMÉN-LINDELÖF THEORY
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 3 , September 1980 , pp. 689 - 709
Copyright
Copyright © Applied Probability Trust 1980 

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References

1. Kagan, A. M., Linnik, Yu. V. and Rao, C. R. (1972) Characterization Problems of Mathematical Statistics. Academy Nauk, Moscow.Google Scholar
2. Lukacs, E. (1969) A characterization of stable process. J. Appl. Prob. 6, 409–418.CrossRefGoogle Scholar
3. Lukacs, E. (1970) Characterization theorems for certain stochastic processes. Rev. Internat. Statist. Inst. 38, 333–343.CrossRefGoogle Scholar
4. Lukacs, E. (1970) Characteristic Functions, 2nd edn. Griffin, London.Google Scholar
5. Lukacs, E. (1975) Stochastic Convergence, 2nd edn. Academic Press, New York.Google Scholar
6. Pólya, G. and Szegö, G. (1925) Aufgaben und Lehrsätze aus der Analysis. Springer-Verlag, Berlin.Google Scholar
7. Riedel, M. (1976) On the onesided tails of infinitely divisible distributions. Math. Nachr. 70, 155–163.CrossRefGoogle Scholar
8. Riedel, M. (1976) Über eine Funktionalgleichung und ihre Anwendung auf Charakterisierungsprobleme von Verteilungsfunktionen aus der Bedienungstheorie und der Statistik. Ph. D. Thesis, Leipzig.Google Scholar
9. Riedel, M. (1977) Some theorems of the Phragmén-Lindelöf theory for subharmonic functions. Math. Nachr. 78, 13–20.CrossRefGoogle Scholar
10. Riedel, M. (1980) Representation of the characteristic function of a stochastic integral. J. Appl. Prob. 17, 448–455.CrossRefGoogle Scholar
11. Robbins, H. (1948) Mixture of distributions. Ann. Math. Statist. 19, 360–369.CrossRefGoogle Scholar
12. Rossberg, H.-J. (1968) Eigenschaften der charakteristischen Funktionen von einseitig beschränkten Verteilungsfunktionen und ihre Anwendung auf ein Charakterisierungsproblem der mathematischen Statistik. Math. Nachr. 37, 37–57.CrossRefGoogle Scholar
13. Rossberg, H.-J. (1974) On a problem of Kolmogorov concerning the normal distribution. Theor. Prob. Appl. 19, 824–828.Google Scholar
14. Rossberg, H.-J. (1975) An extension of the Phragmén-Lindelöf theory which is relevant for characterization theory. In A Modern Course on Statistical Distributions in Scientific Work, Vol. 3, ed. Patil, G. P., Kotz, S., and Ord, J. K. Reidel, Dordrecht.Google Scholar
15. Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton, NJ.Google Scholar