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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.

##### MSC:
 60G50 Sums of independent random variables; random walks 60E99 Distribution theory 62E99 Statistical distribution theory
##### Keywords:
near optimal bounds
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