Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities

ForS=∑ xiξi, where (ξi) is a sequence of independent, symmetric random variables and (xi) is a sequence of vectors in a normed space we give two methods of proving inequalities (E ∥S∥p)1/p⩽Cp, q(E ∥S∥q)1/qwith the constantsCp, qindependent of the sequence (xi). The methods depend on using differential inequalities of Poincaré or logarithmic Sobolev type. The obtained constants are usually better than the ones obtained by other methods.