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Sharp Mixing Rates for Markov Chains on General Spaces with Unbounded Random Environments

Attila Lovas, Miklós Rásonyi, Lionel Truquet

TL;DR

The paper develops a general theory for Markov chains in stationary, unbounded random environments that are contractive up to a bounded perturbation, complemented by a non-standard local minorization. Through a two-stage coupling and quenched–annealed analysis, it proves the existence of a limiting law and a stationary solution, with explicit, sharp mixing-rate bounds that depend on the environment's α-mixing rate. The results improve existing rates (up to logarithmic factors) and apply to broad autoregressive-like models with exogenous covariates, including stochastic gradient Langevin dynamics and volatility models. Together, these contributions yield near-optimal, practically relevant mixing guarantees for MCREs across diverse applications in machine learning and finance.

Abstract

We consider Markov chains on general state spaces in stationary random environment which are defined by a random mapping that is contractive up to a bounded perturbation. We prove their convergence to a limiting law, providing convergence rates. We also show that these processes are strongly mixing and estimate their mixing coefficients. Our results significantly extend those available in the literature. In particular, for some additive autoregressive processes with exogenous covariates we achieve mixing rates that are optimal up to logarithmic factors.

Sharp Mixing Rates for Markov Chains on General Spaces with Unbounded Random Environments

TL;DR

The paper develops a general theory for Markov chains in stationary, unbounded random environments that are contractive up to a bounded perturbation, complemented by a non-standard local minorization. Through a two-stage coupling and quenched–annealed analysis, it proves the existence of a limiting law and a stationary solution, with explicit, sharp mixing-rate bounds that depend on the environment's α-mixing rate. The results improve existing rates (up to logarithmic factors) and apply to broad autoregressive-like models with exogenous covariates, including stochastic gradient Langevin dynamics and volatility models. Together, these contributions yield near-optimal, practically relevant mixing guarantees for MCREs across diverse applications in machine learning and finance.

Abstract

We consider Markov chains on general state spaces in stationary random environment which are defined by a random mapping that is contractive up to a bounded perturbation. We prove their convergence to a limiting law, providing convergence rates. We also show that these processes are strongly mixing and estimate their mixing coefficients. Our results significantly extend those available in the literature. In particular, for some additive autoregressive processes with exogenous covariates we achieve mixing rates that are optimal up to logarithmic factors.

Paper Structure

This paper contains 12 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Assumptions and results
  3. Proofs of main results
  4. Preliminary results
  5. Proof of Proposition \ref{['quenched']}
  6. Proof of Theorem \ref{['limit']}
  7. Proof of Theorem \ref{['mixxi']}
  8. Examples
  9. Machine learning with dependent data
  10. Stochastic volatility models
  11. Threshold autoregressive models
  12. A multivariate autoregressive model

Key Result

Proposition 2.6

Suppose that Assumptions as:drift and min hold true. Suppose furthermore that there exists $y_0\in E$ and $\theta\in (0,1)$ such that $\int d^{\theta}\left(y_0,f(y_0,X_0,u)\right)\mathcal{L}(\varepsilon_1)(du)$ has a logarithmic moment.

Theorems & Definitions (29)

  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Proposition 2.13
  • proof
  • Remark 2.14
  • ...and 19 more