Summary
It is proved that f(x) is the density of a generalized Gamma convolution (GGC), then so is C(1+cx) -γ f(x) provided that γ is a nonnegative integer. Some classes of GGC's are introduced.
References
Bondesson, L.: A general result on infinite divisibility. Ann. Probab. 7, 965–979 (1979)
Bondesson, L.: On the infinite divisibility of products of powers of gamma variables. Z. Wahrscheinlichkeitstheor. Verw. Geb. 49, 171–175 (1979)
Bondesson, L.: On generalized gamma and generalized negative binomial convolutions, Parts I and II. Scand. Actuarial J. 1979, 125–166
Bondesson, L.: Classes of infinitely divisible distributions and densities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 57, 39–71 (1981)
Bondesson, L.: New results on generalized gamma convolutions and the B-class. Scand. Actuarial J. 1984, 197–209
Donoghue, W.F.: Monotone matrix functions and analytic continuation. Berlin Heidelberg New York: Springer 1974
Feller, W.: An introduction to probability theory and its applications, vol. II, 2nd edn. New York: Wiley 1971
Fuchs, B.A., Shabat, B.V.: Functions of a complex variable and some of their applications, vol. I. Oxford: Pergamon Press 1964
Kent, J.: Convolution mixtures of infinitely divisible distributions. Math. Proc. Camb. Phil. Soc. 90, 141–153 (1981)
Thorin, O.: On the infinite divisibility of the Pareto distribution. Scand. Actuarial J. 1977, 31–40
Thorin, O.: On the infinite divisibility of the longormal distribution. Scand. Actuarial J. 1977, 121–148
Thorin, O.: Proof of a conjecture of L. Bondesson concerning infinite divisibility of powers of a gamma variable. Scan. Actuarial J. 1978, 151–164
Yamazato, M.: Unimodality of infinitely divisible distribution functions of class L. Ann. Probab. 6, 523–531 (1978)
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Bondesson, L. A remarkable property of generalized gamma convolutions. Probab. Th. Rel. Fields 78, 321–333 (1988). https://doi.org/10.1007/BF00334198
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DOI: https://doi.org/10.1007/BF00334198
Keywords
- Stochastic Process
- Convolution
- Probability Theory
- Statistical Theory
- Nonnegative Integer