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Maximal Inequalities for Empirical Processes under General Mixing Conditions with an Application to Strong Approximations

Demian Pouzo

TL;DR

The paper develops a maximal inequality for empirical processes indexed by a function class under highly general mixing conditions, allowing arbitrarily slow decay of dependence. It introduces two core ingredients: a dependence measure based on $ au$ and $ heta$-mixing and a Talagrand-style complexity measure that operates over a family of norms tied to block-lengths, enabling a bound of the form $\|\sup_{f,f_0} |G_n[f-f_0]|\|_1 \le L_{a,b}(\gamma_{1,b}/\sqrt{n} + \gamma_{2,a})$ that adapts to the mixing rate. Special cases recover familiar root-$n$ rates under fast mixing, while slow mixing yields slower, phase-transition-type rates with explicit dependence on the mixing structure, and the results extend to Glivenko–Cantelli-type bounds and strong Gaussian approximations without fast-decay assumptions. The framework supports strong approximations by constructing a Gaussian surrogate process and bounding the $L^1$ distance from the empirical process, providing a broad tool for dependent empirical process theory and applications to strong approximations. Overall, the work broadens the empirical process toolkit to dependent data with minimal assumptions on mixing decay, using a flexible complexity measure and a block-based, dependence-adjusted chaining strategy.

Abstract

This paper provides a bound for the supremum of sample averages over a class of functions for a general class of mixing stochastic processes with arbitrary mixing rates. Regardless of the speed of mixing, the bound is comprised of a concentration rate and a novel measure of complexity. The speed of mixing, however, affects the former quantity implying a phase transition. Fast mixing leads to the standard root-n concentration rate, while slow mixing leads to a slower concentration rate, its speed depends on the mixing structure. Our findings are applied to obtain new Glivenko-Cantelli type results and to derive strong approximation results for a general class of mixing processes with arbitrary mixing rates.

Maximal Inequalities for Empirical Processes under General Mixing Conditions with an Application to Strong Approximations

TL;DR

The paper develops a maximal inequality for empirical processes indexed by a function class under highly general mixing conditions, allowing arbitrarily slow decay of dependence. It introduces two core ingredients: a dependence measure based on and -mixing and a Talagrand-style complexity measure that operates over a family of norms tied to block-lengths, enabling a bound of the form that adapts to the mixing rate. Special cases recover familiar root- rates under fast mixing, while slow mixing yields slower, phase-transition-type rates with explicit dependence on the mixing structure, and the results extend to Glivenko–Cantelli-type bounds and strong Gaussian approximations without fast-decay assumptions. The framework supports strong approximations by constructing a Gaussian surrogate process and bounding the distance from the empirical process, providing a broad tool for dependent empirical process theory and applications to strong approximations. Overall, the work broadens the empirical process toolkit to dependent data with minimal assumptions on mixing decay, using a flexible complexity measure and a block-based, dependence-adjusted chaining strategy.

Abstract

This paper provides a bound for the supremum of sample averages over a class of functions for a general class of mixing stochastic processes with arbitrary mixing rates. Regardless of the speed of mixing, the bound is comprised of a concentration rate and a novel measure of complexity. The speed of mixing, however, affects the former quantity implying a phase transition. Fast mixing leads to the standard root-n concentration rate, while slow mixing leads to a slower concentration rate, its speed depends on the mixing structure. Our findings are applied to obtain new Glivenko-Cantelli type results and to derive strong approximation results for a general class of mixing processes with arbitrary mixing rates.
Paper Structure (30 sections, 20 theorems, 123 equations)

This paper contains 30 sections, 20 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. A Maximal $L^{1}(P)$ Inequality
  3. Measure of Dependence.
  4. Measure of Complexity.
  5. Notion of distance.
  6. Choice of block length.
  7. Main Result.
  8. Interpretation and Implications of Theorem \ref{['thm:equi']}
  9. Mixing structure and Geometric structure
  10. The $m$-dependent case
  11. The $\sum_{q=1}^{\infty} \theta(q) < \infty$ case
  12. Maximal Inequality for slow mixing processes
  13. Application to Strong Approximations
  14. Proofs
  15. Proof of Theorem \ref{['thm:equi']}
  16. ...and 15 more sections

Key Result

Theorem 1

For any orders $a,b \geq 1$, there exists a constant $\mathbb{L}_{a,b}$, such that for any $n \in \mathbb{N}$,The constant $\mathbb{L}_{a,b}$ depends on the orders $a,b$ but is otherwise universal; in particular it does not depend on $P,\mathcal{F},$ or $n$. Its exact expression can be found in the

Theorems & Definitions (48)

  • Definition 1: Talagrand's complexity measure
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • proof : Proof of Proposition \ref{['pro:bound.tau.l1']}
  • Example 1
  • Proposition 2
  • proof : Proof of Proposition \ref{['pro:bound.slow.gaussian']}
  • ...and 38 more