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An improvement of Hoffmann-Jørgensen’s inequality. (English) Zbl 1044.60010

Summary: Let \(B\) be a Banach space and \({\mathcal F}\) any family of bounded linear functionals on \(B\) of norm at most one. For \(x\in B\) set \(\| x\|=\sup_{ \Lambda\in {\mathcal F}}\Lambda (x)\) \((\|\cdot\|\) is at least a seminorm on \(B)\). We give probability estimates for the tail probability of \(S_n^*=\max_{1\leq k\leq n}\|\sum^k_{j=1}X_j\|\) where \(\{X_i\}^n_{i=1}\) 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 \(\sum^k_{j=1} Y^{(j)}\), where \(Y^{(r)}\) denotes the \(r\)th largest term of \(\{\| X_i\| \}^n_{i=1}\). 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 \({\mathcal L}^p\) 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 \(p\to\infty\).

MSC:

60E15 Inequalities; stochastic orderings
60B11 Probability theory on linear topological spaces
60G50 Sums of independent random variables; random walks
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