Bernoulli

  • Bernoulli
  • Volume 23, Number 1 (2017), 773-787.

A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions

Anita Behme and Lennart Bondesson

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Abstract

Let $k>0$ be an integer and $Y$ a standard $\operatorname{Gamma}(k)$ distributed random variable. Let $X$ be an independent positive random variable with a density that is hyperbolically monotone (HM) of order $k$. Then $Y\cdot X$ and $Y/X$ both have distributions that are generalized gamma convolutions ($\mathrm{GGC}$s). This result extends a result of Roynette et al. from 2009 who treated the case $k=1$ but without use of the $\mathrm{HM}$-concept. Applications in excursion theory of diffusions and in the theory of exponential functionals of Lévy processes are mentioned.

Article information

Source
Bernoulli Volume 23, Number 1 (2017), 773-787.

Dates
Received: February 2015
Revised: July 2015
First available in Project Euclid: 27 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1475001374

Digital Object Identifier
doi:10.3150/15-BEJ761

Mathematical Reviews number (MathSciNet)
MR3556792

Zentralblatt MATH identifier
1372.60014

Keywords
excursion theory exponential functionals generalized gamma convolution hyperbolic monotonicity Lévy process products and ratios of independent random variables

Citation

Behme, Anita; Bondesson, Lennart. A class of scale mixtures of $\operatorname{Gamma}(k)$-distributions that are generalized gamma convolutions. Bernoulli 23 (2017), no. 1, 773--787. doi:10.3150/15-BEJ761. https://projecteuclid.org/euclid.bj/1475001374


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References

  • [1] Barndorff-Nielsen, O.E., Maejima, M. and Sato, K.-I. (2006). Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12 1–33.
  • [2] Behme, A. and Lindner, A. (2015). On exponential functionals of Lévy processes. J. Theoret. Probab. 28 681–720.
  • [3] Behme, A., Maejima, M., Matsui, M. and Sakuma, N. (2012). Distributions of exponential integrals of independent increment processes related to generalized gamma convolutions. Bernoulli 18 1172–1187.
  • [4] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [5] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191–212.
  • [6] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76. New York: Springer.
  • [7] Bondesson, L. (1997). On hyperbolically monotone densities. In Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz (N. L. Johnson andN. Balakrishnan, eds.) Wiley Ser. Probab. Statist. Appl. Probab. Statist. 299–313. New York: Wiley.
  • [8] Bondesson, L. (2015). A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. J. Theoret. Probab. 28 1063–1081.
  • [9] Bosch, P. and Simon, T. (2014). A proof of Bondesson’s conjecture on stable densities. To appear. Available at arXiv:1411.3369.
  • [10] Bosch, P. and Simon, T. (2015). On the infinite divisibility of inverse Beta distributions. Bernoulli 21 2552–2568.
  • [11] Carmona, P., Petit, F. and Yor, M. (1997). On the distribution and asymptotic results for exponential functionals of Lévy processes. In Exponential Functionals and Principal Values Related to Brownian Motion. Bibl. Rev. Mat. Iberoamericana 73–130. Madrid: Rev. Mat. Iberoamericana.
  • [12] Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc.
  • [13] James, L.F., Roynette, B. and Yor, M. (2008). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 346–415.
  • [14] Jedidi, W. and Simon, T. (2013). Further examples of GGC and HCM densities. Bernoulli 19 1818–1838.
  • [15] Kristiansen, G.K. (1994). A proof of Steutel’s conjecture. Ann. Probab. 22 442–452.
  • [16] Pardo, J.C., Patie, P. and Savov, M. (2012). A Wiener-Hopf type factorization for the exponential functional of Lévy processes. J. Lond. Math. Soc. 86 930–956.
  • [17] Roynette, B., Vallois, P. and Yor, M. (2009). A family of generalized gamma convoluted variables. Probab. Math. Statist. 29 181–204.
  • [18] Salminen, P., Vallois, P. and Yor, M. (2007). On the excursion theory for linear diffusions. Jpn. J. Math. 2 97–127.
  • [19] Sato, K.-I. (2013). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [20] Schilling, R.L., Song, R. and Vondraček, Z. (2012). Bernstein Functions, 2nd ed. de Gruyter Studies in Mathematics 37. Berlin: de Gruyter.
  • [21] Steutel, F.W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Monographs and Textbooks in Pure and Applied Mathematics 259. New York: Dekker, Inc.
  • [22] Thorin, O. (1977). On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977 121–148.
  • [23] Williamson, R.E. (1956). Multiply monotone functions and their Laplace transforms. Duke Math. J. 23 189–207.