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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 41, Fasc. 1,
pages 1 - 7
DOI: 10.37190/0208-4147.41.1.1
Published online 1.4.2021
 

On mixtures of gamma distributions, distributions with hyperbolically monotone densities and generalized gamma convolutions (GGC)} hyperbolically monotone densities and generalized gamma convolutions (GGC)

Tord Sjödin

Abstract: Let \(Y\) be a standard \({\rm Gamma}(k)\) distributed random variable (rv), \(k>0\), and let \(X\) be an independent positive rv. If \(X\) has a hyperbolically monotone density of order \(k\) (\({\rm HM}_k\)), then \(Y\cdot X\) and \(Y/X\) are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with \(Y\) replaced by a finite sum of independent gamma variables with sum of shape parameters at most \(k\). Both results are applied to subclasses of GGC.

2010 AMS Mathematics Subject Classification: Primary 60E10; Secondary 62E15.

Keywords and phrases: gamma distribution, hyperbolically monotone function, Laplace transform, generalized gamma convolution (GGC).

A. Behme and L. Bondesson, A class of scale mixtures of gamma(k)-distributions that are generalized gamma convolutions, Bernoulli 23 (2017), 773–787.

L. Bondesson, Generalized Gamma Convolutions and Related Classes of Distributions and Densities, Lecture Notes in Statist. 76, Springer, New York, 1992.

L. Bondesson, A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables, J. Theoret. Probab. 28 (2015), 1063–1081.

W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley, New York, 1966.

B. Roynette, P. Vallois, and M. Yor, A family of generalized gamma convoluted variables, Probab. Math. Statist. 29 (2009), 181–204.

R. L. Schilling, R. Song, and Z. Vondracek, Bernstein Functions, de Gruyter Stud. Math. 37, de Gruyter, Berlin, 2010.

F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Dekker, New York, 2004.

O. Thorin, On the infinite divisibility of the lognormal distribution, Scand. Actuarial J. 1977, 121–148.

O. Thorin, An extension of the notion of a generalized Γ-convolution, Scand. Actuarial J. 1978, 141–149.

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