Optimal constants in the Rosenthal inequality for random variables with zero odd moments

摘要

We obtain estimates for the best constant in the Rosenthal inequality E vertical bar Sigma(n)(i=1)xi(i)vertical bar(2m) <= C(2m) max(Sigma(n)(i=1) E xi(2m)(i), (Sigma(n)(i=1) E xi(2)(i))(m)) for independent random variables xi(1),...,xi(n) with l zero first odd moments, l >= 1. The estimates are sharp in the extremal cases l = 1 and l = m, that is, in the cases of random variables with zero mean and random variables with m zero first odd moments.