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Maximal Inequalities for Separately Exchangeable Empirical Processes

Harold D. Chiang

TL;DR

This work addresses the gap in maximal inequalities for empirical processes indexed by general function classes when the data are drawn from separately exchangeable arrays with fixed dimension $K$. The authors introduce a transversal-group partition technique to decouple multiway dependencies and employ the Aldous–Hoover–Kallenberg representation to derive both global and local maximal inequalities. The global result provides a bound on the $q$-th moment of the supremum via a uniform entropy integral $J_{\bm{e}}(\delta)$ and an envelope $F$, while the local result delivers sharper control of the first absolute moment through partitioning and Rademacher chaos arguments, with a VC-type refinement for VC-type classes. These contributions extend maximal inequality theory beyond i.i.d. and two-way settings, offering new tools for inference in multiway clustering and networked data contexts with SE structure.

Abstract

This paper derives new maximal inequalities for empirical processes associated with separately exchangeable random arrays. For fixed index dimension $K\ge 1$, we establish a global maximal inequality bounding the $q$-th moment ($q\in[1,\infty)$) of the supremum of these processes. We also obtain a refined local maximal inequality controlling the first absolute moment of the supremum. Both results are proved for a general pointwise measurable function class. Our approach uses a new technique partitioning the index set into transversal groups, decoupling dependencies and enabling more sophisticated higher moment bounds.

Maximal Inequalities for Separately Exchangeable Empirical Processes

TL;DR

This work addresses the gap in maximal inequalities for empirical processes indexed by general function classes when the data are drawn from separately exchangeable arrays with fixed dimension . The authors introduce a transversal-group partition technique to decouple multiway dependencies and employ the Aldous–Hoover–Kallenberg representation to derive both global and local maximal inequalities. The global result provides a bound on the -th moment of the supremum via a uniform entropy integral and an envelope , while the local result delivers sharper control of the first absolute moment through partitioning and Rademacher chaos arguments, with a VC-type refinement for VC-type classes. These contributions extend maximal inequality theory beyond i.i.d. and two-way settings, offering new tools for inference in multiway clustering and networked data contexts with SE structure.

Abstract

This paper derives new maximal inequalities for empirical processes associated with separately exchangeable random arrays. For fixed index dimension , we establish a global maximal inequality bounding the -th moment () of the supremum of these processes. We also obtain a refined local maximal inequality controlling the first absolute moment of the supremum. Both results are proved for a general pointwise measurable function class. Our approach uses a new technique partitioning the index set into transversal groups, decoupling dependencies and enabling more sophisticated higher moment bounds.

Paper Structure

This paper contains 5 sections, 7 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results
  4. Conclusion
  5. Auxiliary Lemmas

Key Result

Theorem 1

Suppose $\mathcal{F}:\mathcal{S}\to \mathbb{R}$ is a pointwise measurable class of functions with an envelope $F$. Let $(X_{\bm{i}})_{{\bm{i}}\in [\bm{N}]}$ be a sample from $S$-valued separately exchangeable random vectors $(X_{\bm{i}})_{{\bm{i}}\in \mathbb{N}^K}$. Pick any $1 \le k \le K$ and $\bm

Theorems & Definitions (10)

  • Theorem 1: Global maximal inequality for SE processes
  • proof
  • Theorem 2: Local maximal inequality for SE processes
  • Remark 1
  • proof
  • Lemma 1: Partitioning into transversal groups
  • Corollary 1
  • Lemma 2: Local maximal inequality under i.i.d.
  • Lemma 3: Bounding $L^q$-norm by Orlicz norm
  • Lemma 4: Properties of $J_{\bm{e}}(\delta)$