| 1. |
- Bondesson, Lennart, 1944-
(författare)
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A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables
- 2015
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Ingår i: Journal of theoretical probability. - : Springer. - 0894-9840 .- 1572-9230. ; 28:3, s. 1063-1081
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Tidskriftsartikel (refereegranskat)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 R, has many consequences and applications. In particular, it follows that X∼ GGC implies that exp(X)∼ GGC. The latter result is used to find a large class of explicit probability functions on {0,1,2,…} which are generalized negative binomial convolutions (GNBCs). The paper ends with several open problems.
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| 2. |
- Bondesson, Lennart, 1944-, et al.
(författare)
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Stieltjes functions of finite order and hyperbolic monotonicity
- 2018
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Ingår i: Transactions of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9947 .- 1088-6850. ; 370:6, s. 4201-4222
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Tidskriftsartikel (refereegranskat)abstract
- A class of Stieltjes functions of finite type is introduced. These satisfy Widder's conditions on the successive derivatives up to some finite order and are not necessarily smooth. We show that such functions have a unique integral representation along some generic kernel which is a truncated Laurent series approximating the standard Stieltjes kernel. We then obtain a two-to-one correspondence, via the logarithmic derivative, between these functions and a subclass of hyperbolically monotone functions of finite type. This correspondence generalizes a representation of HCM functions in terms of two Stieltjes transforms earlier obtained by the first author.
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