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Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes

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  • Mengsi Gao
  • Demian Pouzo

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.

Suggested Citation

  • Mengsi Gao & Demian Pouzo, 2026. "Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes," Papers 2606.31936, arXiv.org.
    • Handle: RePEc:arx:papers:2606.31936
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    References listed on IDEAS

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