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December, 1979 A General Result on Infinite Divisibility
Lennart Bondesson
Ann. Probab. 7(6): 965-979 (December, 1979). DOI: 10.1214/aop/1176994890


Using and refining a technique developed by O. Thorin, we prove: THEOREM. Let $f(x) = C\cdot x^{\beta - 1} h(x), x > 0$, be a probability density on $(0, \infty)$. Here $\beta > 0$ and $h$ is continuous and satisfies $h(0) = 1$. Assume that $h$ can be analytically continued to the whole complex plane cut along the negative real axis and assume that $h$ satisfies some other regularity assumptions. If $h$ is completely monotone on $(0, \infty)$ and if, for each fixed $u > 0$, the function $h(u\nu(t))h(u/\nu(t))$, where $\nu(t) = t + 1 + (t^2 + 2t)^{\frac{1}{2}}$, is completely monotone on $(0, \infty)$, then $f(x)$ is the density of a generalized gamma convolution and hence infinitely divisible. The theorem is applied to show the infinite divisibility of a rather large class of probability densities on $(0, \infty)$. In particular we show that a power with exponent of modulus $\geqslant 1$ of the ratio of two gamma distributed rv's has an infinitely divisible distribution.


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Lennart Bondesson. "A General Result on Infinite Divisibility." Ann. Probab. 7 (6) 965 - 979, December, 1979.


Published: December, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0421.60014
MathSciNet: MR548891
Digital Object Identifier: 10.1214/aop/1176994890

Primary: 60E05

Keywords: Analytic function , complete monotonicity , generalized gamma convolution , Infinite divisibility , moment generating function , Stieltjes transform

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 6 • December, 1979
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