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Non-uniform Berry-Esseen theorems for weakly dependent random variables

Yeor Hafouta

TL;DR

The article develops non-uniform Berry-Esseen bounds for weakly dependent sums, covering broad classes such as inhomogeneous Markov chains, nonstationary dynamical systems, and products of random matrices. It introduces an abstract framework governed by GrowAssum, derives Edgeworth-type expansions via cumulant polynomials, and proves sharp $O(\sigma_n^{-1})$ convergence with polynomially decaying tails. The results yield L^p Gaussian estimates for distribution functions and Gaussian-type bounds for expectations of functions with polynomially growing derivatives, enabling application to stationary and nonstationary settings alike. Overall, the work extends non-uniform BE theory to dependent structures with slow variance growth, using spectral perturbation and cumulant machinery to obtain precise, widely applicable rates and their consequences for statistics of complex systems.

Abstract

We obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics.

Non-uniform Berry-Esseen theorems for weakly dependent random variables

TL;DR

The article develops non-uniform Berry-Esseen bounds for weakly dependent sums, covering broad classes such as inhomogeneous Markov chains, nonstationary dynamical systems, and products of random matrices. It introduces an abstract framework governed by GrowAssum, derives Edgeworth-type expansions via cumulant polynomials, and proves sharp convergence with polynomially decaying tails. The results yield L^p Gaussian estimates for distribution functions and Gaussian-type bounds for expectations of functions with polynomially growing derivatives, enabling application to stationary and nonstationary settings alike. Overall, the work extends non-uniform BE theory to dependent structures with slow variance growth, using spectral perturbation and cumulant machinery to obtain precise, widely applicable rates and their consequences for statistics of complex systems.

Abstract

We obtain non-uniform Berry-Esseen type estimates for several classes of weakly dependent sequences of random variables, including uniformly elliptic inhomogeneous Markov chains, random and time-varying (partially) hyperbolic or expanding dynamical systems, products of random matrices and some classes of local statistics.
Paper Structure (34 sections, 23 theorems, 174 equations)

This paper contains 34 sections, 23 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Berry-Esseen theorems
  3. Main results: non-uniform Berry-Esseen theorems
  4. Two applications of the non-uniform estimates
  5. $L^p$ type Gaussian estimates of distribution function
  6. Gaussian type estimates of expectations of functions of $W_n$ with polynomially fast growing derivatives
  7. Main abstract result
  8. Gaussian estimates of functions with polynomially fast growing derivatives
  9. Proofs of the main abstract results
  10. "Reduction" to the self normalized case: proof of Theorem \ref{['BE2']} relying on the self-normalized case
  11. The Edgeworth polynomials and their Fourier transforms
  12. Proof of Theorem \ref{['BE2']} in the self normalized case
  13. Proof of Theorems \ref{['BE2']} relying on Propositions \ref{['Prp17.1']} and \ref{['Prp3']}
  14. Proof of the main Propositions \ref{['Prp17.1']} and \ref{['Prp3']}
  15. Proof of Proposition \ref{['Prp3']}
  16. ...and 19 more sections

Key Result

Theorem 4

Under Assumption GrowAssum, the non-uniform Berry-Esseen theorem with power $s=m$ holds true, namely for all $x\in{\mathbb R}$ we have where $C$ is some constant. In particular, for every $p>1/m$ we have

Theorems & Definitions (51)

  • Definition 1
  • Remark 3
  • Theorem 4
  • Corollary 5
  • Proposition 6
  • Proposition 7
  • Lemma 8
  • proof
  • proof : Proof of Theorem \ref{['BE2']}
  • proof : Proof of Proposition \ref{['Prp17.1']}
  • ...and 41 more