A two-parameter extension of urbanik’s product convolution semigroup
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A two-parameter extension of urbanik’s product convolution semigroup. / Berg, Christian.
In: Probability and Mathematical Statistics, Vol. 39, No. 2, 2019, p. 441-458.Research output: Contribution to journal › Journal article › Research › peer-review
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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 -
ID: 234561762