## The Annals of Probability

### Limit Theorems for $U$-Processes

#### 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

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
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