On the infinite divisibility of inverse beta distributions. (English) Zbl 1362.60012

Bosch, Pierre; Simon, Thomas

Bernoulli 21, No. 4, 2552-2568 (2015).

Summary: We show that all negative powers \(\beta_{a,b}^{-s}\) of the beta distribution are infinitely divisible. The case \(b\leq1\) follows by complete monotonicity, the case \(b>1\), \(s\geq 1\) 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 \(2a+b+s+bs\geq 1\), 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.

MSC:

60E07	Infinitely divisible distributions; stable distributions
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

beta distribution; gamma distribution; generalized gamma convolution; hyperbolically complete monotonicity; hypergeometric series; Lévy perpetuity; self-decomposability; Stieltjes transform

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