Some results concerning maximum Rényi entropy distributions
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...
A natural derivative on [0, n] and a binomial Poincaré inequality
We consider probability measures supported on a finite discrete interval [0, ]. We introduce a new finite difference operator ∇, defined as a linear combination of left and right finite differences. We show that this operator ∇ plays a key role in a new Poincaré (spectral gap) inequality with respect to binomial weights, with the orthogonal Krawtchouk polynomials acting as eigenfunctions of the relevant operator. We briefly discuss the relationship of this operator to the problem of optimal transport...
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