Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes

2026-06-30

AI Summary

This work establishes novel maximal inequalities for empirical processes under graph-dependent observations, revealing that their convergence rates are jointly governed by the complexity of the function class, the underlying graph structure, and the decay of dependence. By integrating graph coloring, block partitioning, and graph-adapted dependence coefficients, the authors develop a coupling strategy that effectively disentangles the complexity of the indexing class from the graph-induced dependence, thereby overcoming the classical √n convergence barrier. The theoretical framework accommodates graphs with polynomial or exponential growth as well as directed binary networks, yielding Glivenko–Cantelli-type results and explicit quantifications of effective sample size. These advances enable uniform laws of large numbers in applications such as network autoregressive models, nonlinear local propagation dynamics, and settings involving treatment interference.

Abstract

We develop maximal inequalities for empirical processes indexed by graph-dependent observations. Our bounds separate the complexity of the indexing class from two features specific to graph dependence: the geometry of the underlying graph and the cost of coupling graph-separated blocks to independent copies. The coupling construction combines a novel graph-adapted dependence coefficient with a coloring of a block partition. We specialize the results to graphs with polynomial and exponential growth and to directed dyadic graphs. We then derive Glivenko--Cantelli results and characterize the associated effective sample size. A central implication is that graph-dependent empirical processes need not exhibit a generic root-$n$ rate: convergence is jointly determined by function-class complexity, graph geometry, and the decay of dependence with graph distance. Finally, we apply the results to obtain uniform laws of large numbers for network autoregressive models, nonlinear local-propagation models, and treatment-interference settings.

Problem

• graph-dependent empirical processes
• maximal inequalities
• graph geometry
• dependence decay
• effective sample size

Innovation

• graph-dependent empirical processes
• maximal inequalities
• coupling construction
• dependence coefficient
• effective sample size