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Sharper bounds for Gaussian and empirical processes. (English) Zbl 0798.60051
Let \((\Omega, \Sigma,P)\) be a probability space, \(f\) a real-valued random variable on \(\Omega\), and \(X_ 1, \dots, X_ n\) i.i.d. \(\Omega\)-valued random variables on \(\Omega\) with common law \(P\). For certain classes \(F\) of random variables \(f\), near optimal bounds are given for the probabilities \(P(\sup_{f \in F} | \sum_{i \leq n} f(X_ i)-nE(f) | \geq M \sqrt n)\). The basic idea of the author’s approach is borrowed from a previous paper of him [Ann. Inst. Henri Poincaré, Probab. Stat. 24, No. 2, 307-315 (1988; Zbl 0641.60044)]. An impressive machinery is set up, which is important in its own right.

60G50 Sums of independent random variables; random walks
60E99 Distribution theory
62E99 Statistical distribution theory
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