Research Article

Preservation of log-concavity on summation

Johnson, Oliver^a1a2 and Goldschmidt, Christina^a1a3

^a1 Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge, CB3 0WB, UK.

^a2 Christ's College, Cambridge; otj1000@cam.ac.uk

^a3 Pembroke College, Cambridge; C.Goldschmidt@statslab.cam.ac.uk

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.

(Received April 15 2005)

(Online publication May 3 2006)

Key Words:

• Log-concavity;
• convolution;
• dependent random variables;
• Stirling numbers;
• Eulerian numbers.

Mathematics Subject Classification:

• 60E15;
• 60C05;
• 11B75