Sharper bounds for Gaussian and empirical processes. (English) Zbl 0798.60051

Talagrand, M.

Ann. Probab. 22, No. 1, 28-76 (1994).

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.

Reviewer: M.Iosifescu (Bucureşti)

MSC:

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

Keywords:

near optimal bounds

Citations:

Zbl 0641.60044