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dependent randomvariables

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Preservation of log-concavity on summation
Oliver Johnson, Christina Goldschmidt
Journal:

ESAIM: Probability and Statistics / Volume 10 / February 2006

Published online by Cambridge University Press:

03 May 2006, pp. 206-215

Print publication:

February 2006
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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.