Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-uniform Edgeworth expansions for weakly dependent random variables and their applications

Yeor Hafouta

TL;DR

This work addresses non-uniform Edgeworth expansions for weakly dependent, non-stationary sequences, extending classical results beyond the i.i.d. setting to chaotic dynamics, Markov chains, and products of random matrices. By introducing an abstract scheme with Growth and Derivative assumptions (GrowAssum and DerAss), it derives explicit Edgeworth polynomials and tight error bounds that survive non-stationarity and dependence. The paper then develops a self-normalized framework, provides explicit formulas for Edgeworth corrections, and establishes higher-order corrections in both distribution and transport metrics, including L^p Gaussian estimates and Wasserstein BE-type results. Collectively, these results enable precise distributional approximations and moment expansions for a broad class of weakly dependent systems with practical implications for dynamical systems, random matrices, and sequential stochastic processes.

Abstract

We obtain non-uniform Edgeworth expansions for several classes of weakly dependent (non-stationary) 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. To the best of our knowledge this is the first time such results are obtained beyond the case of independent summands, even for stationary sequences. As an application of the non uniform expansions we obtain average versions of Edgeworth exapnsions, which provide estimates of the underlying distribution function in $L^p(dx)$ by the standard normal distribution function and its higher order corrections. An additional application is to expansions of expectations $\bbE[h(S_n)]$ of functions $h$ of the underlying sequence $S_n$, whose derivatives grow at most polynomially fast. In particular we provide expansions of the moments of $S_n$ by means the variance of $S_n$. A third application is to Edgeworth expansions in the Wasserstein distance (transport distance). In particular we prove Berry-Esseen theorems in the Wasserstein metrics. This paper compliments \cite{NonU BE} where non-uniform Berry-Esseen theorems were obtained.

Non-uniform Edgeworth expansions for weakly dependent random variables and their applications

TL;DR

This work addresses non-uniform Edgeworth expansions for weakly dependent, non-stationary sequences, extending classical results beyond the i.i.d. setting to chaotic dynamics, Markov chains, and products of random matrices. By introducing an abstract scheme with Growth and Derivative assumptions (GrowAssum and DerAss), it derives explicit Edgeworth polynomials and tight error bounds that survive non-stationarity and dependence. The paper then develops a self-normalized framework, provides explicit formulas for Edgeworth corrections, and establishes higher-order corrections in both distribution and transport metrics, including L^p Gaussian estimates and Wasserstein BE-type results. Collectively, these results enable precise distributional approximations and moment expansions for a broad class of weakly dependent systems with practical implications for dynamical systems, random matrices, and sequential stochastic processes.

Abstract

We obtain non-uniform Edgeworth expansions for several classes of weakly dependent (non-stationary) 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. To the best of our knowledge this is the first time such results are obtained beyond the case of independent summands, even for stationary sequences. As an application of the non uniform expansions we obtain average versions of Edgeworth exapnsions, which provide estimates of the underlying distribution function in by the standard normal distribution function and its higher order corrections. An additional application is to expansions of expectations of functions of the underlying sequence , whose derivatives grow at most polynomially fast. In particular we provide expansions of the moments of by means the variance of . A third application is to Edgeworth expansions in the Wasserstein distance (transport distance). In particular we prove Berry-Esseen theorems in the Wasserstein metrics. This paper compliments \cite{NonU BE} where non-uniform Berry-Esseen theorems were obtained.

Paper Structure

This paper contains 45 sections, 30 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Berry-Esseen theorems and Edgeworth expansions
  3. Main results: non-uniform Edgeworth expansions
  4. Applications of the non-uniform estimates
  5. $L^p$ type Gaussian estimates of distribution function
  6. Expansions of functions of $W_n$ with polynomially fast growing derivatives
  7. Transport distances
  8. Main abstract result
  9. An explicit formula for the polynomials in the general self-normalized case
  10. Expansions of functions with polynomially fast growing derivatives
  11. Better than optimal CLT rates in transport distances
  12. Applications to Gaussian coupling: A Berry-Esseen theorem in $L^p$
  13. Reduction to the case $A_n=0$ and $B_n={\sigma}_n$: proof of Theorem \ref{['Edge2']} relying on the self-normalized case
  14. Proofs of the main results in the self normalized case
  15. The Edgeworth polynomials
  16. ...and 30 more sections

Key Result

Theorem 8

Let Assumptions GrowAssum and DerAss hold with the same $m$. Then there are polynomials $H_{j,n}$ with bounded coefficients whose degrees do not depend on $n$ so that with for all $r=1,2,...,m-2$ and $x\in{\mathbb R}$ we haveNamely, the non-uniform Edgeworth expansion or order $r$ and power $m$ holds true. In particular, for all $p>1/m$, The coefficients of the polynomials are polynomial functio

Theorems & Definitions (68)

  • Definition 1: Non Uniform Exapnsions
  • Definition 2: Stationary Non Uniform Exapnsions
  • Remark 3
  • Remark 5
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • Corollary 11
  • Remark 12
  • ...and 58 more