On a conjecture of Lamkin and Tkocz: Log-convexity of moments of Bernoulli sample means

Abstract

Let $X_1,X_2,\ldots$ be independent $\mathrm{Bernoulli}(θ)$ random variables, and let $\bar X_n = n^{-1}(X_1 + \cdots + X_n)$. We prove that, for every real $p \geq 1$, the sequence $\{\mathsf{E}(\bar X_n^p)\}_{n \geq 1}$ is log-convex. This settles the Bernoulli case of a conjecture of Lamkin and Tkocz [Canad. Math. Bull., 65(2):271-278, 2022]. The proof conditions on the total number of successes among $2n$ trials and reduces the desired inequality to a convex-order comparison for a normalized quadratic function of a hypergeometric random variable. The log-convexity inequality is strict for $p > 1$ and $0 < θ< 1$.