An improvement of Hoffmann-Jørgensen’s inequality

Abstract

Let $B$ be a Banach space and $\mathscr{F}$ any family of bounded linear functionals on $B$ of norm at most one. For $x\inB\set \|x\| = \sup_{\Lambda \in \mathscr{F}} \Lambda(x)(\|\cdot\|$ is at least a seminorm on $B$). We give probability estimates for the tail probability of $S_n^*= \max_{1 \leq k \leq n}\|\sum_{j=1}^{k} X_j\|$ where $\{X_i\}_{i=1}^{n}$ are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that $S_n^*$ exceeds a threshold defined in terms of $\sum_{j=1}^k Y^{(j)}$, where $Y^{(r)}$ denotes the $r$th largest term of $\{\|X_i\|\}_{i=1}^n$. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as $E\Phi(\|S_n|\|)$ and $E\Phi(S_n^*)$ follow (for any fixed $1 \leq n \leq \infty)$. Included in this paper are uniform $\mathscr{L}^p$ bounds of $S_n^*$ which are within a factor of 4 for all $p \geq 1$ and within a factor of 2 in the limit as $p \to \infty$.