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Statistical properties of Markov shifts (part I)

Yeor Hafouta

TL;DR

The paper develops a comprehensive framework to establish central limit theorems, Berry-Esseen bounds, almost sure invariance principles, and large deviations for sums of path-dependent functionals $S_n= o -1 f_j(...,X_{j-1},X_j,X_{j+1},...)$ driven by inhomogeneous Markov chains. It introduces a robust mix of mixing/approximation conditions, block-decomposition arguments, Sinai-type reductions, and martingale coboundary representations to handle nonstationarity and dependence on the entire chain, yielding optimal CLT rates and LDPs in both stationary and random environment settings. The results apply broadly, including products of random matrices, random Lyapunov exponents, linear processes, iterated random functions, and Hölder-on-average observables in dynamical systems, with a companion paper addressing local limit theorems. By unifying operator perturbation theory, variance regularity (Livsic theory), and probabilistic limit theorems in sequential and random environments, the work significantly extends classical CLT theory to nonautonomous and path-dependent contexts with broad applicability.

Abstract

We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form $S_n=\sum_{j=0}^{n-1}f_j(...,X_{j-1},X_j,X_{j+1},...)$, where $(X_j)$ is an inhomogeneous Markov chain satisfying some mixing assumptions and $f_j$ is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions $f_j$ is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when $(X_j)$ are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}.

Statistical properties of Markov shifts (part I)

TL;DR

The paper develops a comprehensive framework to establish central limit theorems, Berry-Esseen bounds, almost sure invariance principles, and large deviations for sums of path-dependent functionals driven by inhomogeneous Markov chains. It introduces a robust mix of mixing/approximation conditions, block-decomposition arguments, Sinai-type reductions, and martingale coboundary representations to handle nonstationarity and dependence on the entire chain, yielding optimal CLT rates and LDPs in both stationary and random environment settings. The results apply broadly, including products of random matrices, random Lyapunov exponents, linear processes, iterated random functions, and Hölder-on-average observables in dynamical systems, with a companion paper addressing local limit theorems. By unifying operator perturbation theory, variance regularity (Livsic theory), and probabilistic limit theorems in sequential and random environments, the work significantly extends classical CLT theory to nonautonomous and path-dependent contexts with broad applicability.

Abstract

We prove central limit theorems, Berry-Esseen type theorems, almost sure invariance principles, large deviations and Livsic type regularity for partial sums of the form , where is an inhomogeneous Markov chain satisfying some mixing assumptions and is a sequence of sufficiently regular functions. Even though the case of non-stationary chains and time dependent functions is more challenging, our results seem to be new already for stationary Markov chains. They also seem to be new for non-stationary Bernoulli shifts (that is when are independent but not identically distributed). This paper is the first one in a series of two papers. In \cite{Work} we will prove local limit theorems including developing the related reduction theory in the sense of \cite{DolgHaf LLT, DS}.

Paper Structure

This paper contains 56 sections, 41 theorems, 353 equations.

Table of Contents

  1. Introduction
  2. CLT rates for independent summands
  3. Weakly dependent summands
  4. Stationary sequences
  5. Nonstationary sequences
  6. Sequential dynamical systems and non-stationary processes
  7. Random dynamical systems
  8. Preliminaries and main results
  9. Mixing and approximation conditions
  10. Moments and mixing type assumptions needed for optimal CLT rates
  11. A moment condition for block decompositions
  12. The growth of the variance, Livsic regularity and the CLT
  13. Optimal CLT rates and moment estimates
  14. Large and moderate deviations
  15. Moderate deviations
  16. ...and 41 more sections

Key Result

Proposition 2.15

Suppose $\lim_{n\to\infty}{\sigma}_n=\infty$. Then Assumption Stand MomAss holds in the following cases: (1) if mix holds with $p=\infty$ and some $1\leq q\leq p$ and $\sup_j\|f_j\|_{j,\infty,\infty,\delta}<\infty$ for some $\delta\in(0,1)$ the Assumption Stand MomAss holds with every finite $k_0$. where $G_\ell=f_\ell^2-{\mathbb E}[f_\ell^2]$ and $u$ is the conjugate exponent of $p$. (3) Assumpt

Theorems & Definitions (81)

  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.8
  • Definition 2.9
  • Remark 2.13
  • Remark 2.14
  • Proposition 2.15
  • Theorem 2.16
  • Remark 2.17
  • ...and 71 more