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Statistical properties of Markov shifts: part II-LLT

Yeor Hafouta

TL;DR

This work extends local limit theorems to partial sums of equicontinuous, path-dependent observables along non-stationary Markov shifts. By developing a non-autonomous transfer-operator framework (sequential Perron–Frobenius theory) with tailored norms and a Lasota–Yorke inequality, the authors establish non-lattice LLTs and lattice LLTs (via reduction) and first-order Edgeworth corrections without assuming variance linear growth. A reduction mechanism, Sinai-type decomposition for two-sided observables, and a block-structure analysis underpin the irreducible/reducible dichotomy, clarifying obstructions to LLT (lattice, summability, coboundary). The results apply to a broad class of inhomogeneous Markov processes and have concrete implications for products of random matrices, random Lyapunov exponents, linear processes, and iterated random functions in random environments.

Abstract

We prove Local Central Limit Theorems (LLT) 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 a Markov chains with equicontinuous conditional probabilities satisfying contraction conditions close in spirit to Dobrushin's, and some ``physicality" assumptions and $f_j$ are equicontinuous functions. Our conditions will always be in force when the chain takes values on a metric space and have uniformly bounded away from $0$ backward transition densities with respect to a measure which assigns uniform positive mass to certain ``balls". This paper complements \cite{MarShif1} where Berry-Esseen theorems, were proven for (not necessarily continuous) functions satisfying certain approximation conditions. Our results address a question posed by D. Dolgopyat and O. Sarig in \cite[Section 1.5]{DS}.

Statistical properties of Markov shifts: part II-LLT

TL;DR

This work extends local limit theorems to partial sums of equicontinuous, path-dependent observables along non-stationary Markov shifts. By developing a non-autonomous transfer-operator framework (sequential Perron–Frobenius theory) with tailored norms and a Lasota–Yorke inequality, the authors establish non-lattice LLTs and lattice LLTs (via reduction) and first-order Edgeworth corrections without assuming variance linear growth. A reduction mechanism, Sinai-type decomposition for two-sided observables, and a block-structure analysis underpin the irreducible/reducible dichotomy, clarifying obstructions to LLT (lattice, summability, coboundary). The results apply to a broad class of inhomogeneous Markov processes and have concrete implications for products of random matrices, random Lyapunov exponents, linear processes, and iterated random functions in random environments.

Abstract

We prove Local Central Limit Theorems (LLT) for partial sums of the form , where is a Markov chains with equicontinuous conditional probabilities satisfying contraction conditions close in spirit to Dobrushin's, and some ``physicality" assumptions and are equicontinuous functions. Our conditions will always be in force when the chain takes values on a metric space and have uniformly bounded away from backward transition densities with respect to a measure which assigns uniform positive mass to certain ``balls". This paper complements \cite{MarShif1} where Berry-Esseen theorems, were proven for (not necessarily continuous) functions satisfying certain approximation conditions. Our results address a question posed by D. Dolgopyat and O. Sarig in \cite[Section 1.5]{DS}.

Paper Structure

This paper contains 39 sections, 42 theorems, 229 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Preliminaries and main results
  4. The setup and standing assumptions
  5. On the behavior of the variance and the CLT
  6. Main results: reduction theory, local limit theorems and first order Edgeworth expansions
  7. An overview: obstructions to LLT
  8. Some immediate consequences of our standing assumptions
  9. Non-autonomous functional analytic tools
  10. Norms of functions
  11. A sequential Perron Frobenius theorem
  12. Norms of complex transfer operators
  13. A complex Perron-Frobenius theorem
  14. Lasota Yorke inequality and related results
  15. Integral of characteristic function and LLT
  16. ...and 24 more sections

Key Result

Theorem 2.9

Under Assumptions MixAss and Assss the following conditions are equivalent. (1) $\liminf_{n\to\infty}\text{Var}(S_nf)<\infty$; (2) $\sup_{n\in{\mathbb N}}\text{Var}(S_n f)<\infty$; (3) we can write where $\sup_j\|u_j\|_{j,{\omega}}<\infty,$$\sup_j\|M_j\|_{j,{\omega}}\!\!<\!\!\infty$, $u_j$ and $M_j$ have zero mean and $M_j(X_j,X_{j+1},...),j\geq 0$ is a reverse martingale difference with respec

Theorems & Definitions (80)

  • Remark 2.2
  • Remark 2.5
  • Example 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 2.9
  • Remark 2.10
  • Definition 2.11
  • Remark 2.12
  • Theorem 2.13
  • ...and 70 more