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Effective geometric ergodicty for Markov chains in random environment

Yeor Hafouta

TL;DR

The paper proves effective geometric ergodicity for Markov chains in random environments under a random Doeblin condition, producing a random invariant family $μ_ω$ and exponential convergence with rate $ρ∈(0,1)$ tempered by integrable random constants $K_p(ω)$. The rate and constants are explicitly constructed via a regenerative scheme and upper mixing bounds, ensuring $K_p∈L^p$ for any $p≥1$. This yields quantitative quenched ASIP rates, exponential decay of correlations for Markovian skew products, and exponential tails for random mixing times, with additional verifiable conditions for existing results (e.g., Kifer 1998). The framework connects probabilistic ergodicity with spectral-type interpretations in random operator cocycles and extends prior non-effective results to an effective, integrable setting. These results provide practical, verifiable conditions and rates for limit theorems in random dynamical systems and related stochastic processes.

Abstract

In this short note we prove ``effective" geometric ergodicity (i.e a Perron-Frobenius theorem) for Markov chains in random mixing dynamical environment satisfying a random non-uniform version of the Doeblin condition. Effectivity here means that all the random variables involved in the random exponential rates are integrable with arbitrarily large order. This compliments \cite[Theorem 2.1]{Kifer 1996}, where ``non-effective" geometric ergodicity was obtained. From a different perspective, our result is also motivated by egrodic theory, as it can be seen as an effective version of the ``spectral" gap in the top Oseledets space in the Oseledets multiplicative ergodic theorem for the random Markov operator cocycle (when it applies). We also present applications of the effective ergodicity to rates in the (quenched) almost sure invariance principle (ASIP), exponential decay of correlations for Markovian skew products and for exponential tails for random mixing times. As a byproduct of the proof of the ASIP rates we also provide easy to verify sufficient conditions for the verification of the assumptions of \cite[Theorem 2.4]{Kifer 1998}.

Effective geometric ergodicty for Markov chains in random environment

TL;DR

The paper proves effective geometric ergodicity for Markov chains in random environments under a random Doeblin condition, producing a random invariant family and exponential convergence with rate tempered by integrable random constants . The rate and constants are explicitly constructed via a regenerative scheme and upper mixing bounds, ensuring for any . This yields quantitative quenched ASIP rates, exponential decay of correlations for Markovian skew products, and exponential tails for random mixing times, with additional verifiable conditions for existing results (e.g., Kifer 1998). The framework connects probabilistic ergodicity with spectral-type interpretations in random operator cocycles and extends prior non-effective results to an effective, integrable setting. These results provide practical, verifiable conditions and rates for limit theorems in random dynamical systems and related stochastic processes.

Abstract

In this short note we prove ``effective" geometric ergodicity (i.e a Perron-Frobenius theorem) for Markov chains in random mixing dynamical environment satisfying a random non-uniform version of the Doeblin condition. Effectivity here means that all the random variables involved in the random exponential rates are integrable with arbitrarily large order. This compliments \cite[Theorem 2.1]{Kifer 1996}, where ``non-effective" geometric ergodicity was obtained. From a different perspective, our result is also motivated by egrodic theory, as it can be seen as an effective version of the ``spectral" gap in the top Oseledets space in the Oseledets multiplicative ergodic theorem for the random Markov operator cocycle (when it applies). We also present applications of the effective ergodicity to rates in the (quenched) almost sure invariance principle (ASIP), exponential decay of correlations for Markovian skew products and for exponential tails for random mixing times. As a byproduct of the proof of the ASIP rates we also provide easy to verify sufficient conditions for the verification of the assumptions of \cite[Theorem 2.4]{Kifer 1998}.
Paper Structure (9 sections, 6 theorems, 63 equations)

This paper contains 9 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and effective geometric ergodicity
  3. Proof of the effective geometric ergodicity (Theorem \ref{['Main']})
  4. A key lemma
  5. Proof of Theorem \ref{['Main']}
  6. Applications
  7. Application to quenched ASIP rates
  8. Application to exponential decay of correlations for skew products
  9. Applications to random mixing times

Key Result

Theorem 2.2

[Effective random geometric ergodicity] Under Assumption AssMix and DoebRand there exists a unique random family of measures $\mu_{\omega}$ such that $(R_\omega)^*\mu_{\omega}=\mu_{{\theta}{\omega}}$, for ${\mathbb P}$-a.a. ${\omega}$. Moreover, there exists $\rho\in(0,1)$ such that for every finite

Theorems & Definitions (14)

  • Theorem 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 4 more