Document 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, 31 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