• 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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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.

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

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

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


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

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