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A simple lemma concerning the Doeblin minorization condition and its applications to limit theorems for inhomogeneous Markov chains

Yeor Hafouta, Brenden Williams

TL;DR

The work provides an elementary bridge from nonuniform sequential Doeblin minorization to nonuniform geometric ergodicity for inhomogeneous Markov chains, unlocking limit theorems with explicit rates under relatively mild nonuniformity. By developing a martingale-coboundary decomposition and precise minorization propagation lemmas, it yields CLTs, Berry-Esseen-type bounds, and Wasserstein convergence in both purely sequential and random dynamical environments. The authors introduce effective polynomial and stretched exponential mixing rates, derive variance limits, and extend results to skew-product settings, with applications to mixing times and correlation decay. This framework offers practical criteria for obtaining quantitative limit theorems in non-stationary stochastic systems without requiring uniform ellipticity.

Abstract

In this short note we provide an elementary proof that a certain type of nonuniform sequential Doeblin minorization condition implies non-uniform sequential "geometric" ergodicity. Using this result several limit theorems for inhomogeneous Markov chains follow immediately from existing results [7,8,12]. We then focus our attention to Markov chains in random dynamical environment and deduce effective mixing rates which imply limit theorems for such processes by using the methods of [16] (which were formulated in a dynamical setup). The crucial part of the proof is to obtain effective convergence rates towards the random equivariant distribution, which has its own interest and yields, for instance, effective mixing times estimates together with results for the corresponding skew products

A simple lemma concerning the Doeblin minorization condition and its applications to limit theorems for inhomogeneous Markov chains

TL;DR

The work provides an elementary bridge from nonuniform sequential Doeblin minorization to nonuniform geometric ergodicity for inhomogeneous Markov chains, unlocking limit theorems with explicit rates under relatively mild nonuniformity. By developing a martingale-coboundary decomposition and precise minorization propagation lemmas, it yields CLTs, Berry-Esseen-type bounds, and Wasserstein convergence in both purely sequential and random dynamical environments. The authors introduce effective polynomial and stretched exponential mixing rates, derive variance limits, and extend results to skew-product settings, with applications to mixing times and correlation decay. This framework offers practical criteria for obtaining quantitative limit theorems in non-stationary stochastic systems without requiring uniform ellipticity.

Abstract

In this short note we provide an elementary proof that a certain type of nonuniform sequential Doeblin minorization condition implies non-uniform sequential "geometric" ergodicity. Using this result several limit theorems for inhomogeneous Markov chains follow immediately from existing results [7,8,12]. We then focus our attention to Markov chains in random dynamical environment and deduce effective mixing rates which imply limit theorems for such processes by using the methods of [16] (which were formulated in a dynamical setup). The crucial part of the proof is to obtain effective convergence rates towards the random equivariant distribution, which has its own interest and yields, for instance, effective mixing times estimates together with results for the corresponding skew products
Paper Structure (13 sections, 19 theorems, 72 equations)

This paper contains 13 sections, 19 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Purely sequential setting
  4. Markov chains in random dynamical environment with non-uniform Doeblin minorization
  5. Purely inhomogeneous setting
  6. Mixing rates via Doeblin minorization
  7. A one step minorization
  8. A general sequential minorization
  9. Limit theorems
  10. Proof of Theorem \ref{['VarThm']}
  11. Proof of Theorems \ref{['BE']} and \ref{['ThWass']}
  12. Application to Markov chains in random environment: effective geometric ergodicity and limit theorems
  13. Skew products

Key Result

Theorem 2.1

The following conditions are equivalent: (i) ${\sigma}_n$ is bounded; (ii) $\liminf_{n\to\infty}{\sigma}_n<\infty;$ (iii) we have for some measurable uniformly bounded functions $h_j$ and $M_j$ is a uniformly bounded martingale difference such that $\sum_{j=1}^\infty\text{Var}(M_j(X_{j-1},X_j))<\infty$ (and so $\sum_{j=1}^\infty M_j(X_{j-1},X_j)$ converges almost surely and in $L^p$ for all $1\le

Theorems & Definitions (27)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.5
  • Remark 2.6
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • Theorem 2.11
  • Theorem 2.12
  • ...and 17 more