A TWO-PARAMETER EXTENSION OF URBANIK’S PRODUCTCONVOLUTION SEMIGROUP

Christian Berg

Abstract: We prove that , is an infinitely divisible
Stieltjes moment sequence for arbitrary . Its powers , are Stieltjes
determinate if and only if . The latter was conjectured in a paper by Lin (2019) in the
case . We describe a product convolution semigroup , , of probability
measures on the positive half-line with densities and having the moments
. We determine the asymptotic behavior of for and for
, and the latter implies the Stieltjes indeterminacy when . The results extend
the previous work of the author and López (2015) and lead to a convolution semigroup of
probability densities on the real line. The special case
are the convolution roots of the Gumbel distribution with scale parameter .
All the densities lead to determinate Hamburger moment problems.