# Empirical Process Results for Exchangeable Arrays

## TL;DR

This paper establishes uniform laws of large numbers and central limit theorems for exchangeable arrays, which model dependence in dyadic data and multiway clustering, extending classical results to dependent data structures.

## Contribution

It provides the first uniform laws of large numbers and CLTs for exchangeable arrays, including bootstrap convergence, under conditions similar to i.i.d. data.

## Key findings

- Proves uniform laws of large numbers for exchangeable arrays.
- Establishes central limit theorems for exchangeable arrays.
- Demonstrates bootstrap convergence for dependent array data.

## Abstract

Exchangeable arrays are natural tools 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 those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.