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Asymptotic behaviour of the Urbanik semigroup. (English) Zbl 1317.30058

Summary: We revisit the product convolution semigroup of probability densities \(e_c(t)\), \(c > 0\) on the positive half-line with moments \((n!)^c\) and determine the asymptotic behaviour of \(e_c\) for large and small \(t > 0\). This shows that \((n!)^c\) is indeterminate as Stieltjes moment sequence if and only if \(c > 2\). When \(c\) is a natural number \(e_c\) is a Meijer-G function. From the results about \(e_c\) we obtain the asymptotic behaviour at \(\pm \infty\) of the convolution roots of the Gumbel distribution.

MSC:

30E15 Asymptotic representations in the complex plane
44A60 Moment problems
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization

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References:

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