Preservation of log-concavity on summation

Oliver Johnson; Christina Goldschmidt

ESAIM: Probability and Statistics (2006)

  • Volume: 10, page 206-215
  • ISSN: 1292-8100

Abstract

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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.

How to cite

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Johnson, Oliver, and Goldschmidt, Christina. "Preservation of log-concavity on summation." ESAIM: Probability and Statistics 10 (2006): 206-215. <http://eudml.org/doc/249749>.

@article{Johnson2006,
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. },
author = {Johnson, Oliver, Goldschmidt, Christina},
journal = {ESAIM: Probability and Statistics},
keywords = {Log-concavity; convolution; dependent random variables; Stirling numbers; Eulerian numbers.; log-concavity; dependent random variables; Eulerian numbers},
language = {eng},
month = {5},
pages = {206-215},
publisher = {EDP Sciences},
title = {Preservation of log-concavity on summation},
url = {http://eudml.org/doc/249749},
volume = {10},
year = {2006},
}

TY - JOUR
AU - Johnson, Oliver
AU - Goldschmidt, Christina
TI - Preservation of log-concavity on summation
JO - ESAIM: Probability and Statistics
DA - 2006/5//
PB - EDP Sciences
VL - 10
SP - 206
EP - 215
AB - 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.
LA - eng
KW - Log-concavity; convolution; dependent random variables; Stirling numbers; Eulerian numbers.; log-concavity; dependent random variables; Eulerian numbers
UR - http://eudml.org/doc/249749
ER -

References

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