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On powers of Stieltjes moment sequences, I. (English) Zbl 1086.44003

The starting point of this work is a theorem proved earlier by the author and A. J. Duran [Ark. Mat. 42, 239–257 (2004; Zbl 1057.44002)]. It is proved that the border \(c=2\) in this theorem is best possible by proving that the sequence \((n!)c\) becomes indeterminate as a Stieltjes moment sequence when \(c>2\). The proof is given in section 2. In section 3 the author deals with a similar result about the product of independent identically distributed normal random variables. It is also proved that the distribution of the product of \(p\) independent identically distributed normal random variables is indeterminate if and only if \(p\) is greater than or equal to 3.

MSC:

44A60 Moment problems
60E07 Infinitely divisible distributions; stable distributions

Citations:

Zbl 1057.44002
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References:

[24] Skorokhod A.V. (1954). Asymptotic formulas for stable distribution laws. Dokl. Akad. Nauk SSSR 98, 731–734; English transl. (1961), Selected Transl. Math. Statist. and Probab., Vol. 1, Amer. Math. Soc., Providence, R.I. · Zbl 0057.11106
[26] Stieltjes T.J. (1894,1895). Recherches sur les fractions continues. Annales de la Faculté des Sciences de Toulouse 8, 1–122, 9, 5–47
[31] Zolotarev V.M. (1986). One-dimensional Stable Distributions, Translations of Mathematical Monographs, 65, Amer. Math. Soc., Providence, R.\(\sim\)I. (Russian edition 1983)
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