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Convolution property and exponential bounds for symmetric monotone densities

Published online by Cambridge University Press:  06 August 2013

Claude Lefèvre  and
Sergey Utev
Claude Lefèvre
Affiliation:
Département de Mathématique, Université Libre de Bruxelles, Campus de la Plaine C.P. 210, 1050 Bruxelles, Belgique. clefevre@ulb.ac.be
Sergey Utev
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, NG7 2RD Nottingham, UK; sergey.utev@nottingham.ac.uk
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Abstract

Our first theorem states that the convolution of two symmetric densities which are k-monotone on (0,∞) is again (symmetric) k-monotone provided 0 < k ≤ 1. We then apply this result, together with an extremality approach, to derive sharp moment and exponential bounds for distributions having such shape constrained densities.

Keywords

Multiply monotonicitysymmetric densitiesunimodalityWintner’s theoremBernstein’s inequality
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 605 - 613
Copyright
© EDP Sciences, SMAI, 2013

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