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A Two-Parameter Extension of Urbanik’s Product Convolution Semigroup

Berg Christian

Probability and Mathematical Statistics

2019

Vol. 39, Fasc. 2

441--458

We prove that sn(a, b) = Γ(an + b)/Γ(b), n = 0, 1,…, is an infinitely divisible Stieltjes moment sequence for arbitrary a, b > 0. Its powers sn (a, b)c, c > 0, are Stieltjes determinate if and only if ac ≤ 2. The latter was conjectured in a paper by Lin (2019) in the case b = 1. We describe a product convolution semigroup τc(a, b), c > 0, of probability measures on the positive half-line with densities ec (a, b) and having the moments sn (a, b)c. We determine the asymptotic behavior of ec (a, b)(t) for t → 0 and for t → ∞, and the latter implies the Stieltjes indeterminacy when ac > 2. The results extend the previous work of the author and López (2015) and lead to a convolution semigroup of probability densities (gc (a, b)(x))c>0 on the real line. The special case (gc (a, 1)(x))c>0 are the convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gc (a, b)(x) lead to determinate Hamburger moment problems.