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 $f^{\prime \prime }(y)$ is convex, and all independent centered random variables $\xi$ and $\eta$ 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 $A_{\rho }$ and $B_{\rho }$ 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 $\rho \ge 1$, $\xi$ and $\eta$ are independent centered random variables. We write out similar sharp inequalities for symmetric random variables.

arXiv commentThe article is accepted for publication in Statistics & Probability Letters

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