Bernoulli

  • Bernoulli
  • Volume 21, Number 4 (2015), 2552-2568.

On the infinite divisibility of inverse Beta distributions

Pierre Bosch and Thomas Simon

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Abstract

We show that all negative powers $\beta_{a,b}^{-s}$ of the Beta distribution are infinitely divisible. The case $b\le1$ follows by complete monotonicity, the case $b>1$, $s\ge1$ by hyperbolically complete monotonicity and the case $b>1$, $s<1$ by a Lévy perpetuity argument involving the hypergeometric series. We also observe that $\beta_{a,b}^{-s}$ is self-decomposable if and only if $2a+b+s+bs\ge1$, and that in this case it is not necessarily a generalized Gamma convolution. On the other hand, we prove that all negative powers of the Gamma distribution are generalized Gamma convolutions, answering to a recent question of L. Bondesson.

Article information

Source
Bernoulli, Volume 21, Number 4 (2015), 2552-2568.

Dates
Received: May 2014
First available in Project Euclid: 5 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1438777604

Digital Object Identifier
doi:10.3150/14-BEJ654

Mathematical Reviews number (MathSciNet)
MR3378477

Zentralblatt MATH identifier
1362.60012

Keywords
Beta distribution Gamma distribution generalized Gamma convolution hyperbolically complete monotonicity hypergeometric series Lévy perpetuity self-decomposability Stieltjes transform

Citation

Bosch, Pierre; Simon, Thomas. On the infinite divisibility of inverse Beta distributions. Bernoulli 21 (2015), no. 4, 2552--2568. doi:10.3150/14-BEJ654. https://projecteuclid.org/euclid.bj/1438777604


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References

  • [1] Anderson, G.D., Vamanamurthy, M.K. and Vuorinen, M. (2007). Generalized convexity and inequalities. J. Math. Anal. Appl. 335 1294–1308.
  • [2] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [3] Bertoin, J. and Yor, M. (2002). On the entire moments of self-similar Markov processes and exponential functionals of Lévy processes. Ann. Fac. Sci. Toulouse Math. (6) 11 33–45.
  • [4] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191–212.
  • [5] Bondesson, L. (1992). Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics 76. New York: Springer.
  • [6] Bondesson, L. (2014). A class of probability distributions that is closed with respect to addition as well as multiplication of independent random variables. J. Theoret. Probab. To appear. Available at DOI:10.1007/s10959-013-0523-y.
  • [7] Bosch, P. (2014). HCM property and the half-Cauchy distribution. Available at arXiv:1402.1059.
  • [8] Bosch, P. and Simon, T. (2013). On the self-decomposability of the Fréchet distribution. Indag. Math. (N.S.) 24 626–636.
  • [9] Bustoz, J. and Ismail, M.E.H. (1986). On gamma function inequalities. Math. Comp. 47 659–667.
  • [10] Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G. (1953). Higher Transcendental Functions Vol. I and II. New York: McGraw-Hill.
  • [11] Gjessing, H.K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stochastic Process. Appl. 71 123–144.
  • [12] James, L.F., Roynette, B. and Yor, M. (2008). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv. 5 346–415.
  • [13] Janson, S. (2010). Moments of gamma type and the Brownian supremum process area. Probab. Surv. 7 1–52.
  • [14] Jedidi, W. and Simon, T. (2013). Further examples of GGC and HCM densities. Bernoulli 19 1818–1838.
  • [15] Klein, F. (1890). Ueber die Nullstellen der hypergeometrischen Reihe. Math. Ann. 37 573–590.
  • [16] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. Translated from the 1990 Japanese original. Revised by the author.
  • [17] Schilling, R.L., Song, R. and Vondraček, Z. (2010). Bernstein Functions. Theory and Applications. De Gruyter Studies in Mathematics 37. Berlin: de Gruyter.
  • [18] Simon, T. (2014). Comparing Fréchet and positive stable laws. Electron. J. Probab. 19 1–25.
  • [19] Steutel, F.W. and Van Harn, K. (2003). Infinite Divisibility of Probability Distributions on the Real Line. New York: Dekker.
  • [20] Van Vleck, E.B. (1902). A determination of the number of real and imaginary roots of the hypergeometric series. Trans. Amer. Math. Soc. 3 110–131.
  • [21] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. in Appl. Probab. 11 750–783.