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

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  • Lennart Bondesson

    (Umeå University)

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.

Suggested Citation

  • Lennart Bondesson, 2015. "A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables," Journal of Theoretical Probability, Springer, vol. 28(3), pages 1063-1081, September.
    • Handle: RePEc:spr:jotpro:v:28:y:2015:i:3:d:10.1007_s10959-013-0523-y
    DOI: 10.1007/s10959-013-0523-y
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    References listed on IDEAS

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    1. Steutel, F. W., 1973. "Some recent results in infinite divisibility," Stochastic Processes and their Applications, Elsevier, vol. 1(2), pages 125-143, April.
    2. Kozubowski, Tomasz J., 2005. "A note on self-decomposability of stable process subordinated to self-decomposable subordinator," Statistics & Probability Letters, Elsevier, vol. 73(4), pages 343-345, July.
    3. Kozubowski, Tomasz J., 2005. "A note on self-decomposability of stable process subordinated to self-decomposable subordinator," Statistics & Probability Letters, Elsevier, vol. 74(1), pages 89-91, August.
    4. Antonio Lijoi & Ramsés Mena & Igor Prünster, 2005. "Bayesian Nonparametric Analysis for a Generalized Dirichlet Process Prior," Statistical Inference for Stochastic Processes, Springer, vol. 8(3), pages 283-309, December.
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    Cited by:

    1. Anthony G. Pakes, 2020. "Self-Decomposable Laws from Continuous Branching Processes," Journal of Theoretical Probability, Springer, vol. 33(1), pages 361-395, March.
    2. Nuha Altaymani & Wissem Jedidi, 2023. "New Monotonicity and Infinite Divisibility Properties for the Mittag-Leffler Function and for Stable Distributions," Mathematics, MDPI, vol. 11(19), pages 1-26, September.
    3. Higbee, Joshua D. & McDonald, James B., 2024. "A comparison of the GB2 and skewed generalized log-t distributions with an application in finance," Journal of Econometrics, Elsevier, vol. 240(2).

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