Issue |
ESAIM: PS
Volume 10, September 2006
|
|
---|---|---|
Page(s) | 206 - 215 | |
DOI | https://doi.org/10.1051/ps:2006008 | |
Published online | 03 May 2006 |
Preservation of log-concavity on summation
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:
15
April
2005
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
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.