Bernoulli

  • Bernoulli
  • Volume 19, Number 5A (2013), 1818-1838.

Further examples of GGC and HCM densities

Wissem Jedidi and Thomas Simon

Full-text: Open access

Abstract

We display several examples of generalized gamma convoluted and hyperbolically completely monotone random variables related to positive $\alpha$-stable laws. We also obtain new factorizations for the latter, refining Kanter’s and Pestana–Shanbhag–Sreehari’s. These results give stronger credit to Bondesson’s hypothesis that positive $\alpha$-stable densities are hyperbolically completely monotone whenever $\alpha\le1/2$.

Article information

Source
Bernoulli, Volume 19, Number 5A (2013), 1818-1838.

Dates
First available in Project Euclid: 5 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.bj/1383661204

Digital Object Identifier
doi:10.3150/12-BEJ431

Mathematical Reviews number (MathSciNet)
MR3129035

Zentralblatt MATH identifier
1303.60016

Keywords
generalized Gamma convolution hyperbolically completely monotone hyperbolically monotone positive stable density

Citation

Jedidi, Wissem; Simon, Thomas. Further examples of GGC and HCM densities. Bernoulli 19 (2013), no. 5A, 1818--1838. doi:10.3150/12-BEJ431. https://projecteuclid.org/euclid.bj/1383661204


Export citation

References

  • [1] Bondesson, L. (1981). Classes of infinitely divisible distributions and densities. Z. Wahrsch. Verw. Gebiete 57 39–71. Corection and Addendum 59, 277.
  • [2] Bondesson, L. (1990). Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 85 181–194.
  • [3] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76. New York: Springer.
  • [4] Bondesson, L. (1999). A problem concerning stable distributions. Technical report, Uppsala Univ.
  • [5] Bondesson, L. (2009). On univariate and bivariate generalized gamma convolutions. J. Statist. Plann. Inference 139 3759–3765.
  • [6] Chaumont, L. and Yor, M. (2003). Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, via Conditioning. Cambridge Series in Statistical and Probabilistic Mathematics 13. Cambridge: Cambridge Univ. Press.
  • [7] Cuculescu, I. and Theodorescu, R. (1998). Multiplicative strong unimodality. Aust. N. Z. J. Stat. 40 205–214.
  • [8] Demni, N. (2011). Kanter random variable and positive free stable distributions. Electron. Commun. Probab. 16 137–149.
  • [9] Devroye, L. (2009). Random variate generation for exponentially and polynomially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation 19 Article 18.
  • [10] Diédhiou, A. (1998). On the self-decomposability of the half-Cauchy distribution. J. Math. Anal. Appl. 220 42–64.
  • [11] Erdélyi, A. (1953). Higher Transcendental Functions, Vol. III. New York: McGraw-Hill.
  • [12] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. New York: Wiley.
  • [13] James, L.F. (2010). Lamperti-type laws. Ann. Appl. Probab. 20 1303–1340.
  • [14] James, L.F., Roynette, B. and Yor, M. (2008). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 346–415.
  • [15] Kanter, M. (1975). Stable densities under change of scale and total variation inequalities. Ann. Probab. 3 697–707.
  • [16] Lamperti, J. (1958). An occupation time theorem for a class of stochastic processes. Trans. Amer. Math. Soc. 88 380–387.
  • [17] Pillai, R.N. (1990). On Mittag–Leffler functions and related distributions. Ann. Inst. Statist. Math. 42 157–161.
  • [18] Roynette, B., Vallois, P. and Yor, M. (2009). A family of generalized gamma convoluted variables. Probab. Math. Statist. 29 181–204.
  • [19] Satô, K. (1999). 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. (2010). Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics 37. Berlin: de Gruyter.
  • [21] Shanbhag, D.N., Pestana, D. and Sreehari, M. (1977). Some further results in infinite divisibility. Math. Proc. Cambridge Philos. Soc. 82 289–295.
  • [22] Simon, T. (2011). Multiplicative strong unimodality for positive stable laws. Proc. Amer. Math. Soc. 139 2587–2595.
  • [23] Simon, T. (2012). On the unimodality of power transformations of positive stable densities. Math. Nachr. 285 497–506.
  • [24] Steutel, F.W. and Van Harn, K. (2003). Infinite Divisibility of Probability Distributions on the Real Line. New York: Dekker.