Exact Maximal Inequalities for Exchangeable Systems of Random Variables

Abstract

Given an exchangeable finite system of Banachspace valued random variables $(\xi_1,\ldots,\xi_n)$ with $\sum_1^n\xi_i=0$, we prove that {\bf E}$\Phi(\max_{k\le n}\|\xi_1+\cdots+\xi_k\|)$ is equivalent to {\bf E}$\Phi(\|\sum_1^n\xi_ir_i\|)$ for any increasing and convex $\Phi$: {\bf R}$^+\to{\bf R}^+$, $\Phi(0)=0$, where $(r_1,\ldots,r_n)$ is a system of Rademacher random variables independent of $(\xi_1,\ldots,\xi_n)$. We also establish the equivalence of the tails of the related distributions. The results seem to be new also for scalar random variables. As corollaries we find best estimations for the average of $\max_{k\le n}\|a_{\pi(1)}+\cdots+a_{\pi(k)}\|$ with respect to permutations~$\pi$ of nonrandom vectors $a_1,\ldots,a_n$ from a normed space.