On Cox-Kemperman moment inequalities for independent centered random variables

Abstract

In 1983 Cox and Kemperman proved that $\E f(ξ)+ \E f(η) \le \E f(ξ+η)$ for all functions $f$, such that $f(0)=0$ and the second derivative $f''(y)$ is convex, and all independent centered random variables $ξ$ and $η$ satisfying certain moment restrictions. We show that the minimal moment restrictions are sufficient for the inequality to be valid, and write out a less restrictive condition on $f$ for the inequality to hold. Besides, Cox and Kemperman (1983) found out the optimal constants $A_ρ$ and $B_ρ$ for the inequalities $A_ρ(\E |ξ|^ρ+ \E |η|^ρ) \le \E |ξ+ η|^ρ\le B_ρ(\E |ξ|^ρ+ \E |η|^ρ) $, where $ρ\ge1$, $ξ$ and $η$ are independent centered random variables. We write out similar sharp inequalities for symmetric random variables.