A Threshold for Log-Concavity for Probability Generating Functions and Associated Moment Inequalities

Abstract

Let $\{p_n\}_0^N$ be a discrete distribution on $0 \leqq n \leqq N$ and let $g(u) = \sum^\infty_0 p_n u^n$ be its $\operatorname{pgf}$. Then for $0 \leqq t < \infty g_t(u) = g(u + t)/g(1 + t) = \sum^N_0 p_n(t)u^n$ is a family of $\operatorname{pgf}$'s indexed by $t$. It is shown that there is a unique value $t^\ast$ such that $\{p_n(t)\}_0^N$ is $\log$-concave $(PF_2)$ for all $t \geqq t^\ast$ and is not $\log$-concave for $0 < t < t^\ast$. As a consequence one finds the infinite set of moment inequalities $\{\mu_{\lbrack r\rbrack}/\mathbf{r}!\}^{1/r} \geqq \{\mu_{\lbrack r+1\rbrack}/(r + 1)!\}^{1/r+1} \mathbf{r} = 1,2,3,\cdots$ etc. where $\mu_{\lbrack r\rbrack}$ is the $\mathbf{r}$th factorial moment of $\{p_n\}_0^N$ when the lattice distribution is $\log$-concave. The known set of inequalities for the continuous analogue is shown to follow from the discrete inequalities.