1,169 research outputs found
Preservation of log-concavity on summation
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
Interlacing Log-concavity of the Boros-Moll Polynomials
We introduce the notion of interlacing log-concavity of a polynomial sequence
, where is a polynomial of degree m with
positive coefficients . This sequence of polynomials is said to be
interlacing log-concave if the ratios of consecutive coefficients of
interlace the ratios of consecutive coefficients of for any . Interlacing log-concavity is stronger than the log-concavity. We show that
the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a
sufficient condition for interlacing log-concavity which implies that some
classical combinatorial polynomials are interlacing log-concave.Comment: 10 page
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