Probability and Mathematical Statistics: Vol. 39, Fasc. 2

A TWO-PARAMETER EXTENSION OF URBANIK’S PRODUCT CONVOLUTION SEMIGROUP

Christian Berg

Abstract: We prove that $s (a,b) = Γ (an+ b)∕Γ (b),n = 0,1,... n$ , is an infinitely divisible Stieltjes moment sequence for arbitrary $a,b > 0$ . Its powers $s (a,b)c,c > 0 n$ , 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 $τ(a,b) c$ , $c > 0$ , of probability measures on the positive half-line with densities $e(a,b) c$ and having the moments $s (a,b)c n$ . We determine the asymptotic behavior of $e (a,b)(t) c$ 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 $(g (a,b)(x)) c c>0$ on the real line. The special case $(g (a,1)(x)) c c>0$ are the convolution roots of the Gumbel distribution with scale parameter $a > 0$ . All the densities $g (a,b)(x) c$ lead to determinate Hamburger moment problems.

2000 AMS Mathematics Subject Classification: Primary: 60E07; Secondary: 60B15, 44A60.

Keywords and phrases: Infinitely divisible Stieltjes moment sequence, product convolution semigroup, asymptotic approximation of integrals, Gumbel distribution.