Skip to main content
SpringerLink
Log in

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

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

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 \(\mathbb {R}\), has many consequences and applications. In particular, it follows that \( X \sim \) GGC implies that \( \exp (X) \sim \) GGC. The latter result is used to find a large class of explicit probability functions on \(\{0,1,2,\ldots \}\) which are generalized negative binomial convolutions (GNBCs). The paper ends with several open problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barndorff-Nielsen, O.E., Maejima, M., Sato, K.: Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–33 (2006)

    MATH  MathSciNet  Google Scholar 

  2. Behme, A., Maejima, M., Matsui, M., Sakuma, N.: Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions. Bernoulli 18, 1172–1187 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berg, C.: On a generalized gamma convolution related to the q-calculus. In: Ismail, M.E.H., Koelink, Erik (eds.) Developments of Mathematics Vol 13, Theory and Applications of Special Functions. A Volume Dedicated to Mizan Rahman. Springer Science + Business Media Inc, Berlin (2005)

    Google Scholar 

  4. Bondesson, L.: On the infinite divisibility of products of powers of gamma variables. Z. Wahrsch. verw. Gebiete 49, 171–175 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bondesson, L.: Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76. Springer, New York (1992)

    Book  Google Scholar 

  6. Bondesson, L.: On univariate and bivariate generalized gamma convolutions. J. Stat. Plan. Inference 139, 3759–3765 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bondesson, L., Grandell, J., Peetre, J.: The life and work of Olof Thorin (1912–2004). Proc. Estonian Acad. Sci. Phys. Math. 57, 18–25 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bosch, P., Simon, T.: On the self-decomposability of the Fréchet distribution. Indag. Math. 24, 626–636 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Goovaerts, M.J., d’Hooge, L., de Pril, N.: On the infinite divisibility of the product of two \(\varGamma \)-distributed stochastical variables. Appl. Math. Comput. 3, 127–135 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  10. Grigelionis, B.: On subordinated multivariate Gaussian Lévy processes. Acta Appl. Math. 96, 233–246 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gut, A.: On the moment problem. Bernoulli 8, 407–421 (2002)

    MATH  MathSciNet  Google Scholar 

  12. James, L.F., Zhang, Z.: Quantile clocks. Ann. Appl. Probab. 21, 1627–1662 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. James, L.F., Roynette, B., Yor, M.: Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5, 346–415 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kent, J.T., Mohammadzadeh, M., Mosammam, A.M.: The dimple in Gneiting’s spatial-temporal covariance model. Biometrika 98, 489–494 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kozubowski, T.J.: A note on self-decomposability of stable process subordinated to self-decomposable subordinator. Stat. Probab. Lett. 74, 89–91 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kristiansen, G.: A proof of Steutel’s conjecture. Ann. Probab. 22, 442–452 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lijoi, A., Mena, R.H., Prünster, I.: Bayesian nonparametric analysis for a generalized Dirichlet process prior. Stat. Inference Stoch. Process. 8, 283–309 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Marshall, A.W., Olkin, I.: Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families. Springer Series in Statistics, Berlin (2007)

    Google Scholar 

  19. Pakes, A.G.: Structure of Stieltjes classes of moment-equivalent probability laws. J. Math. Anal. Appl. 326, 1268–1290 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  20. Schilling, R.L., Song, R., Vondraček, Z.: Bernstein Functions. Studies in Mathematics 37. Walter de Gruyter, Berlin (2010)

    Google Scholar 

  21. Shanbhag, D.N., Pestana, D., Sreehari, M.: Some further results in infinite divisibility. Math. Proc. Camb. Philos. Soc. 82, 289–295 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  22. Steutel, F.W.: Some recent results in infinite divisibility. Stoch. Process. Appl. 1, 125–143 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  23. Steutel, F.W., van Harn, K.: Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York (2004)

    MATH  Google Scholar 

  24. Thorin, O.: On the infinite divisibility of the Pareto distribution. Scand. Actuar. J. 1977, 31–40 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  25. Thorin, O.: On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, 121–148 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  26. Vervaat, W.: On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Probab. 11, 750–783 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

Two referees are thanked for very kind comments. One of them urged the author to produce the present more complete proof of the main theorem.

Author information

Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 , Umeå, Sweden

    Lennart Bondesson

Authors
  1. Lennart Bondesson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lennart Bondesson.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bondesson, L. A Class of Probability Distributions that is Closed with Respect to Addition as Well as Multiplication of Independent Random Variables. J Theor Probab 28, 1063–1081 (2015). https://doi.org/10.1007/s10959-013-0523-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-013-0523-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation