Inequalities Between the Second and Fourth Moments

Abstract

Let σ 2 be the variance and μ 4 the fourth moment of a symmetric probability distribution. We will prove that for distributions with non-negative characteristic function the inequality μ 4 ≥ 2σ 4 holds and that μ 4 − 2σ 4 if and only if the characteristic function f is given by f(x) = cos2(ax). for some . For symmetric unimodal distributions we have μ 4 ≥ (9/5)σ 4 and μ 4 = (9/5)σ 4 if and only if the characteristic function f is given by f(x) = (sin(ax))/ax, for some . The products of variances of adjoint positive definite densities have a greatest lower bound A. There is a self-adjoint distribution such that σ 4 = Λ. We will prove that for such distributions the equality μ 4 ≤ 2 + σ 4 holds.