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.