A two-parameter extension of urbanik’s product convolution semigroup

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

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>0on the real line. The special case(gc(a, 1)(x)) are the c>0 convolution roots of the Gumbel distribution with scale parameter a > 0. All the densities gc(a, b)(x) lead to determinate Hamburger moment problems.

OriginalsprogEngelsk
TidsskriftProbability and Mathematical Statistics
Vol/bind39
Udgave nummer2
Sider (fra-til)441-458
Antal sider18
ISSN0208-4147
DOI
StatusUdgivet - 2019

Links

ID: 234561762