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Positive stable densities and the bell-shape


Author: Thomas Simon
Journal: Proc. Amer. Math. Soc. 143 (2015), 885-895
MSC (2010): Primary 60E07, 62E15
DOI: https://doi.org/10.1090/S0002-9939-2014-12256-8
Published electronically: October 8, 2014
MathSciNet review: 3283675
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Abstract: We show that positive stable densities are bell-shaped; that is, their $ n$-th derivatives vanish exactly $ n$ times on $ (0,+\infty )$ and have an alternating sign sequence. This confirms the graphic predictions of Holt and Crow (1973) in the positive case.


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Thomas Simon
Affiliation: Laboratoire Paul Painlevé, Université Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Email: simon@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12256-8
Keywords: Bell-shape, exponential mixture, exponential sum, positive stable density, total positivity
Received by editor(s): February 4, 2013
Received by editor(s) in revised form: May 10, 2013
Published electronically: October 8, 2014
Additional Notes: Ce travail a bénéficié d’une aide de l’Agence Nationale de la Recherche portant la référence ANR-09-BLAN-0084-01.
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.