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Preservation of log-concavity on summation

Published online by Cambridge University Press:  03 May 2006

Oliver Johnson  and
Christina Goldschmidt
Oliver Johnson
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WB, UK. Christ's College, Cambridge; otj1000@cam.ac.uk
Christina Goldschmidt
Affiliation:
Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WB, UK. Pembroke College, Cambridge; C.Goldschmidt@statslab.cam.ac.uk
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Abstract

We extend Hoggar's theorem that the sum of two independentdiscrete-valued log-concave random variables is itself log-concave. Weintroduce conditions under which the result still holds for dependentvariables. We argue that these conditions are natural by giving someapplications. Firstly, we use our main theorem to give simple proofsof the log-concavity of the Stirling numbers of the second kind and ofthe Eulerian numbers.Secondly, we prove results concerning the log-concavityof the sum of independent (not necessarily log-concave) randomvariables.

Keywords

Log-concavityconvolutiondependent randomvariablesStirling numbersEulerian numbers.

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 10 , February 2006 , pp. 206 - 215
Copyright
© EDP Sciences, SMAI, 2006

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