A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables

Lennart Bondesson ()
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Lennart Bondesson: Umeå University

Journal of Theoretical Probability, 2015, vol. 28, issue 3, 1063-1081

Abstract: Abstract Thorin’s class of generalized gamma convolutions (GGCs) is closed with respect to change in scale, weak limits, and addition of independent random variables. Here, it is shown that the GGC class also has the remarkable property of being closed with respect to multiplication of independent random variables. This novel result, which has a simple extension to symmetric distributions on $$\mathbb {R}$$ R , has many consequences and applications. In particular, it follows that $$ X \sim $$ X ∼ GGC implies that $$ \exp (X) \sim $$ exp ( X ) ∼ GGC. The latter result is used to find a large class of explicit probability functions on $$\{0,1,2,\ldots \}$$ { 0 , 1 , 2 , … } which are generalized negative binomial convolutions (GNBCs). The paper ends with several open problems.

Keywords: Generalized gamma convolution; Generalized negative binomial convolution; Hyperbolic complete monotonicity; Infinite divisibility; Self-decomposability; 60E07; 60E10 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (3)

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DOI: 10.1007/s10959-013-0523-y

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