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Issue ESAIM: PS
Volume 10, 2006
Page(s) 206 - 215
DOI 10.1051/ps:2006008
Published online 03 May 2006

ESAIM: P&S, April 2006, Vol. 10, pp. 206-215
DOI: 10.1051/ps:2006008

Preservation of log-concavity on summation

Oliver Johnson1, 2 and Christina Goldschmidt1, 3

1  Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WB, UK.
2  Christ's College, Cambridge; otj1000@cam.ac.uk
3  Pembroke College, Cambridge; C.Goldschmidt@statslab.cam.ac.uk


(Received April 15, 2005. / Published online: 3 May 2006)

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.


Mathematics Subject Classification. 60E15, 60C05, 11B75.

Key words: Log-concavity, convolution, dependent random variables, Stirling numbers, Eulerian numbers.


© EDP Sciences, SMAI 2006


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