TY - JOUR

T1 - A two-parameter extension of urbanik’s product convolution semigroup

AU - Berg, Christian

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Asymptotic approximation of integrals

KW - Gumbel distribution

KW - Infinitely divisible Stieltjes moment sequence

KW - Product convolution semigroup

UR - http://www.scopus.com/inward/record.url?scp=85077522117&partnerID=8YFLogxK

U2 - 10.19195/0208-4147.39.2.11

DO - 10.19195/0208-4147.39.2.11

M3 - Journal article

AN - SCOPUS:85077522117

VL - 39

SP - 441

EP - 458

JO - Probability and Mathematical Statistics

JF - Probability and Mathematical Statistics

SN - 0208-4147

IS - 2

ER -