## The Annals of Probability

- Ann. Probab.
- Volume 21, Number 3 (1993), 1494-1542.

### Limit Theorems for $U$-Processes

Miguel A. Arcones and Evarist Gine

#### Abstract

Necessary and sufficient conditions for the law of large numbers and sufficient conditions for the central limit theorem for $U$-processes are given. These conditions are in terms of random metric entropies. The CLT and LLN for VC subgraph classes of functions as well as for classes satisfying bracketing conditions follow as consequences of the general results. In particular, Liu's simplicial depth process satisfies both the LLN and the CLT. Among the techniques used, randomization, decoupling inequalities, integrability of Gaussian and Rademacher chaos and exponential inequalities for $U$-statistics should be mentioned.

#### Article information

**Source**

Ann. Probab., Volume 21, Number 3 (1993), 1494-1542.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176989128

**Digital Object Identifier**

doi:10.1214/aop/1176989128

**Mathematical Reviews number (MathSciNet)**

MR1235426

**Zentralblatt MATH identifier**

0789.60031

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 62E20: Asymptotic distribution theory 60F15: Strong theorems 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

**Keywords**

$U$-process uniform central limit theorem uniform law of large numbers metric entropy

#### Citation

Arcones, Miguel A.; Gine, Evarist. Limit Theorems for $U$-Processes. Ann. Probab. 21 (1993), no. 3, 1494--1542. doi:10.1214/aop/1176989128. https://projecteuclid.org/euclid.aop/1176989128