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A Note on Physical Dependence and Mixing Conditions for Triangular Arrays

Florian Heinrichs

TL;DR

We address the problem of relating weak physical dependence for locally stationary triangular arrays to $β$-mixing. By introducing a regularity framework for marginal/conditional densities and employing a mollification–Wasserstein coupling approach, we derive the explicit bound $β(k) \le C\sqrt{D\,Θ_k}$ with $Θ_k=\sum_{h=k}^{∞} δ_1(G,h)$ and $D=\max\{D_1,D_2\}$, thereby showing that weak physical dependence implies $β$-mixing (and thus strong mixing). The results complement Hill (2025) by establishing the reverse implication under mild smoothness assumptions, and they enable the use of strong-mixing–based inference for a broader class of triangular arrays. The approach hinges on a density-distance control via mollification and a Kantorovich–Rubinstein coupling argument to connect density and distributional dependence. The practical impact lies in providing verifiable conditions under which statistical procedures reliant on strong mixing remain valid for locally stationary, weakly physically dependent sequences.

Abstract

Under mild structural assumptions and regularity conditions on the marginal and conditional densities, an explicit bound on the $β$-mixing coefficients in terms of the physical dependence measure is provided. Consequently, weak physical dependence implies $β$-mixing and strong mixing for triangular arrays, complementing Hill (2025), who proved the converse implication under moment assumptions.

A Note on Physical Dependence and Mixing Conditions for Triangular Arrays

TL;DR

We address the problem of relating weak physical dependence for locally stationary triangular arrays to -mixing. By introducing a regularity framework for marginal/conditional densities and employing a mollification–Wasserstein coupling approach, we derive the explicit bound with and , thereby showing that weak physical dependence implies -mixing (and thus strong mixing). The results complement Hill (2025) by establishing the reverse implication under mild smoothness assumptions, and they enable the use of strong-mixing–based inference for a broader class of triangular arrays. The approach hinges on a density-distance control via mollification and a Kantorovich–Rubinstein coupling argument to connect density and distributional dependence. The practical impact lies in providing verifiable conditions under which statistical procedures reliant on strong mixing remain valid for locally stationary, weakly physically dependent sequences.

Abstract

Under mild structural assumptions and regularity conditions on the marginal and conditional densities, an explicit bound on the -mixing coefficients in terms of the physical dependence measure is provided. Consequently, weak physical dependence implies -mixing and strong mixing for triangular arrays, complementing Hill (2025), who proved the converse implication under moment assumptions.
Paper Structure (4 sections, 3 theorems, 37 equations)

This paper contains 4 sections, 3 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Main Result
  3. Auxiliary Results
  4. Proof of Theorem \ref{['thm:main']}

Key Result

Theorem 1

Let $X$ be a triangular array satisfying Assumptions assump:loc_stat and assump:density. There exists an absolute constant $C\ge 0$, such that for every $k\in\mathbb{N}$ where $D = \max\{D_1, D_2\}$.

Theorems & Definitions (8)

  • Example 1
  • Example 2
  • Theorem 1
  • remark 1
  • Proposition 1: Smoothing error
  • proof
  • Proposition 2: Mollified difference in $W_{1,d}$
  • proof