HCM property and the half-Cauchy distribution

• Autorzy: Bosch P.
• Języki publikacji: EN

Abstrakty

EN

Let Z_α and Z_α be two independent positive α-stable random variables. It is known that (Z_α/Z_α)^α is distributed as the positive branch of a Cauchy random variable with drift. We show that the density of the power transformation (Z_α/Z_α)^β is hyperbolically completely monotone in the sense of Thorin and Bondesson if and only if α ≤ 1/2 and |β| ≥ α/(1−α). This clarifies a conjecture of Bondesson (1992) on positive stable densities.

Słowa kluczowe

EN

half-Cauchy distribution; complete monotonicity; generalized gamma convolution; hyperbolically completely monotonne function; Pick function; positive stable density

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2015
• Tom: Vol. 35, Fasc. 2
• Strony: 191--200
• Opis fizyczny: Bibliogr. 16 poz., wykr.

Twórcy

autor

Bosch P.

• Université Lille 1, Laboratoire Paul Painlevé, Cité Scientifique, F-59655 Villeneuve d’Ascq Cedex

Bibliografia

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• [2] L. Bondesson, On the infinite divisibility of the half-Cauchy and other decreasing densities and probability functions on the nonnegative line, Scand. Actuar. J. (3-4) (1987), pp. 225-247.
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• [8] S. Fourati, α-stable densities are hyperbolically completely monotone for α ϵ (0, 1/4] U [1/3, 1/2], arXiv: 1309.1045, September 2013.
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• [13] T. Simon, Multiplicative strong unimodality for positive stable laws, Proc. Amer. Math. Soc. 139 (7) (2011), pp. 2587-2595.
• [14] O. Thorin, Proof of a conjecture of L. Bondesson concerning infinite divisibility of powers of a gamma variable, Scand. Actuar. J. (3) (1978), pp. 151-164.
• [15] D. V. Widder, The Laplace Transform, Princeton Math. Ser., Vol. 6, Princeton University Press, Princeton, N.J., 1941.
• [16] V. M. Zolotarev, One-dimensional Stable Distributions, Transl. Math. Monogr., Vol. 65, American Mathematical Society, Providence, R.I., 1986.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).