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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 independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of independent (not necessarily log-concave) random variables.

Keywords

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

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