A characterization of self-decomposable probabilities on the half-line
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Summary
It is shown that a probability measure μ on ℝ+ is self-decomposable if and only if for s>0 the sequence determines a probability on ℕ0, that is self-decomposable in the sense of Steutel and van Harn.
$$\left( {\frac{1}{{n!}}\mathop \smallint \limits_0^\infty e^{ - ts} (ts)^n d\mu (t)} \right)_{n \geqq 0} ,$$
Keywords
Stochastic Process Probability Measure Probability Theory Mathematical Biology
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References
- 1.Forst, G.: A Characterization of Potential Kernels on the Positive Half-Line. Z. Wahrscheinlichkeitstheorie und verw. Gebiete 41, 335–340 (1978)Google Scholar
- 2.Loève, M.: Probability Theory I, 4th Edition. Berlin-Heidelberg-New York: Springer 1977Google Scholar
- 3.Steutel, F.W.: Preservation of infinite divisibility under mixing and related topics. Math. Centre Tracts 33. Amsterdam: Math. Cent. 1970Google Scholar
- 4.Steutel, F.W., van Harn, K.: Discrete analogues of self-decomposability and stability. Preprint. Technological University, Eindhoven. The Netherlands, 1978. [To appear in Ann. Probability]Google Scholar
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© Springer-Verlag 1979