## 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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### References:

 [1] de Acosta,A. (1980). Strong exponential integrability of sums of independent B-valued random vectors. Probab. Math. Statist. 1 133-150. · Zbl 0502.60007 [2] Hoffmann-Jørgensen, J. (1974). Sums of independent Banach space valued random variables. Studia Math. 52 159-186. · Zbl 0287.60010 [3] Klass,M. J. (1981). A method of approximating expectations of functions of sum of independent random variables. Ann. Probab. 9 413-428. · Zbl 0463.60023 [4] Latala,R. (1997). Estimation of moments of sums of independent real random variables. Ann. Probab. 25 1502-1513. · Zbl 0885.60011 [5] Montgomery-Smith,S. (1990). Distributions of Rademacher sums. Proc. Amer. Math. Soc. 109 517-522. JSTOR: · Zbl 0696.60013 [6] Talagrand,M. (1988). An isoperimetric theorem on the cube and the Khintchine-Kahane inequalities. Proc. Amer. Math. Soc. 104 905-909. · Zbl 0691.60015 [7] Talagrand,M. (1989). Isoperimetry and integrability of the sum of independent Banach-space valued random variables. Ann. Probab. 17 1546-1570. · Zbl 0692.60016 [8] Yurinskii,V. V. (1974). Exponential bounds for large deviations. Theory Probab. Appl. 19 154-155. · Zbl 0323.60029
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