On the infinite divisibility of scale mixtures of symmetric $\alpha$-stable distributions, $\alpha \in (0,1]$

Volume 90 / 2010

G. Mazurkiewicz Banach Center Publications 90 (2010), 79-82 MSC: 60A10, 60B05, 60E05, 60E07, 60E10. DOI: 10.4064/bc90-0-5

Abstract

The paper contains a new and elementary proof of the fact that if $\alpha \in (0,1]$ then every scale mixture of a symmetric $\alpha$-stable probability measure is infinitely divisible. This property is known to be a consequence of Kelker's result for the Cauchy distribution and some nontrivial properties of completely monotone functions. It is known that this property does not hold for $\alpha =2$. The problem discussed in the paper is still open for $\alpha \in (1,2)$.

Authors

  • G. MazurkiewiczFaculty of Mathematics, Computer Science and Econometrics
    University of Zielona Góra
    Szafrana 4a
    65-516 Zielona Góra, Poland
    e-mail

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