## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 1 (1994), 28-76.

### Sharper Bounds for Gaussian and Empirical Processes

#### Abstract

Under natural conditions on a class $\mathscr{F}$ of functions on a probability space, near optimal bounds are given for the probabilities $P\big(\sup_{f\in\mathscr{F}}|\sum_{i\leq n} f(X_i) - nE(f)| \geq M\sqrt n\big)$. The method is a variation of this author's method to study the tail probability of the supremum of a Gaussian process.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 1 (1994), 28-76.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988847

**Digital Object Identifier**

doi:10.1214/aop/1176988847

**Mathematical Reviews number (MathSciNet)**

MR1258865

**Zentralblatt MATH identifier**

0798.60051

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60E99: None of the above, but in this section 62E99: None of the above, but in this section

**Keywords**

Uniform approximation isoperimetric inequalities tail probabilities

#### Citation

Talagrand, M. Sharper Bounds for Gaussian and Empirical Processes. Ann. Probab. 22 (1994), no. 1, 28--76. doi:10.1214/aop/1176988847. https://projecteuclid.org/euclid.aop/1176988847