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.
Mots clés : log-concavity, convolution, dependent random variables, Stirling numbers, eulerian numbers
@article{PS_2006__10__206_0, author = {Johnson, Oliver and Goldschmidt, Christina}, title = {Preservation of log-concavity on summation}, journal = {ESAIM: Probability and Statistics}, pages = {206--215}, publisher = {EDP-Sciences}, volume = {10}, year = {2006}, doi = {10.1051/ps:2006008}, mrnumber = {2219340}, language = {en}, url = {http://www.numdam.org/articles/10.1051/ps:2006008/} }
TY - JOUR AU - Johnson, Oliver AU - Goldschmidt, Christina TI - Preservation of log-concavity on summation JO - ESAIM: Probability and Statistics PY - 2006 SP - 206 EP - 215 VL - 10 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2006008/ DO - 10.1051/ps:2006008 LA - en ID - PS_2006__10__206_0 ER -
Johnson, Oliver; Goldschmidt, Christina. Preservation of log-concavity on summation. ESAIM: Probability and Statistics, Tome 10 (2006), pp. 206-215. doi : 10.1051/ps:2006008. http://www.numdam.org/articles/10.1051/ps:2006008/
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