A two-parameter extension of urbanik’s product convolution semigroup
Research output: Contribution to journal › Journal article › Research › peer-review
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.
Original language | English |
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Journal | Probability and Mathematical Statistics |
Volume | 39 |
Issue number | 2 |
Pages (from-to) | 441-458 |
Number of pages | 18 |
ISSN | 0208-4147 |
DOIs | |
Publication status | Published - 2019 |
- Asymptotic approximation of integrals, Gumbel distribution, Infinitely divisible Stieltjes moment sequence, Product convolution semigroup
Research areas
Links
- https://arxiv.org/pdf/1802.00993.pdf
Accepted author manuscript
ID: 234561762