A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. (English) Zbl 1375.60053
A generalized gamma convolution is a probability distribution on \([0,\infty)\) which is a weak limit as \(n\to\infty\) of \(\mu_1\ast\dots\ast\mu_n\), where \(\mu_i\) is a gamma distribution, and \(\ast\) denotes the convolution. It is known that the class GGC of generalized gamma convolutions is closed with respect to several natural operations. For instance, if \(X_1\) and \(X_2\) are independent random variables with \(\mathrm{Law}(X_i)\in\mathrm{GGC}\), then \(\mathrm{Law}(X_1+X_2)\in \mathrm{GGC}\).
In the present paper, it is proved for the first time that also \(\mathrm{Law}(X_1X_2)\in\mathrm{GGC}\). Several consequences of this result are discussed, the most notable one being that \(\mathrm{Law}(X)\in \mathrm{GGC}\) entails \(\mathrm{Law}(\exp X)\in\mathrm{GGC}\). The last observation enables one to show that certain, explicitly given discrete distribution is a weak limit of convolutions of negative binomial distributions. The paper closes with a number of open problems.
In the present paper, it is proved for the first time that also \(\mathrm{Law}(X_1X_2)\in\mathrm{GGC}\). Several consequences of this result are discussed, the most notable one being that \(\mathrm{Law}(X)\in \mathrm{GGC}\) entails \(\mathrm{Law}(\exp X)\in\mathrm{GGC}\). The last observation enables one to show that certain, explicitly given discrete distribution is a weak limit of convolutions of negative binomial distributions. The paper closes with a number of open problems.
Reviewer: Alexander Iksanov (Kiev)
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
Keywords:
generalized gamma convolution; generalized negative binomial convolution; hyperbolic complete monotonicity; infinite divisibility; self-decomposabilityReferences:
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