Empirical Process Results for Exchangeable Arrays

Abstract

Exchangeable arrays are natural ways to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g. geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as with i.i.d. data. As a result, convergence and asymptotic normality of nonlinear estimators can be obtained under the same regularity conditions as with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.