arXiv 30 Jun 2026 · Mathematics — Probability
arXiv:2606.31936 · PDF · DOI · OpenAlex · Extracted main text
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.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Demian Pouzo (2026) Maximal inequalities for empirical processes under general mixing conditions self | 1.000 | 10 | 4 | 100% |
| 2 | Talagrand, Michel (2014) Upper and Lower Bounds for Stochastic Processes | 1.000 | 8 | 4 | 100% |
| 3 | Dedecker, Jérôme and Merlevede, Florence (2006) Inequalities for partial sums of Hilbert-valued dependent sequences and applications | 1.000 | 7 | 3 | 100% |
| 4 | van der Vaart, Aad W. and Wellner, Jon (1996) Weak Convergence and Empirical Processes with Applications to Statistics | 0.928 | 5 | 4 | 80% |
| 5 | Dedecker, Jérôme and Prieur, Clémentine (2004) Coupling for $ $-dependent sequences and applications | 0.843 | 3 | 3 | 100% |
| 6 | Doukhan, Paul and Massart, Pascal and Rio, Emmanuel (1995) Invariance principles for absolutely regular empirical processes | 0.737 | 3 | 2 | 100% |
| 7 | Ramon van Handel (2018) Chaining, interpolation and convexity II: The contraction principle | 0.737 | 3 | 2 | 100% |
| 8 | van Handel, Ramon (2018) Chaining, interpolation, and convexity | 0.737 | 3 | 2 | 100% |
| 9 | Talagrand, Michel (2005) The Generic Chaining | 0.737 | 3 | 2 | 100% |
| 10 | Yu, Bin (1994) Rates of convergence for empirical processes of stationary mixing sequences | 0.737 | 3 | 2 | 100% |
Showing the top 10 of 37 scored citations.