{"ID":5443857,"CreatedAt":"2026-07-01T02:07:11.383974684Z","UpdatedAt":"2026-07-03T16:35:57.158869329Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.31936","arxiv_id":"2606.31936","title":"Coupling and Maximal Inequalities for Graph-Dependent Empirical Processes","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.","short_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 coupli...","url_abs":"https://arxiv.org/abs/2606.31936","url_pdf":"https://arxiv.org/pdf/2606.31936v1","authors":"[\"Mengsi Gao\",\"Demian Pouzo\"]","published":"2026-06-30T16:42:30Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"econ.EM\",\"math.ST\"]","methods":"[]","has_code":false}