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Berry-Esseen Bounds for Statistics of Non-Stationary, $φ$-Mixing Random Variables

Brendan Williams, Yeor Hafouta

Abstract

Using a modification of Stein's method, we generalize the results of Bentkus, G{ö}tze, and Tikhomirov \cite{bentkus1997berry} to obtain Berry-Esseen bounds for a broad class of statistics of sequences of $φ$-mixing, non-stationary random variables with polynomial mixing rates. %and linear variance. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary $φ$-mixing random variables, including rates for a general class of processes of $φ$-mixing random variables satisfying an aggregate third moment bound.

Berry-Esseen Bounds for Statistics of Non-Stationary, $φ$-Mixing Random Variables

Abstract

Using a modification of Stein's method, we generalize the results of Bentkus, G{ö}tze, and Tikhomirov \cite{bentkus1997berry} to obtain Berry-Esseen bounds for a broad class of statistics of sequences of -mixing, non-stationary random variables with polynomial mixing rates. %and linear variance. We then consider applications of this theorem to ensure Berry-Esseen rates for various classes of non-stationary -mixing random variables, including rates for a general class of processes of -mixing random variables satisfying an aggregate third moment bound.

Paper Structure

This paper contains 17 sections, 13 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results under linear growth of variance
  3. Statement of Berry-Esseen Theorem for Statistics of Non-Stationary, Weakly Dependent Samples
  4. Assumptions
  5. Proof of Theorem \ref{['Berry-Esseen Bounds for Statistics of Weakly Dependent Samples Proof']}
  6. Notes and Extensions of Theorem \ref{['Berry-Esseen Bounds for Statistics of Weakly Dependent Samples Proof']}
  7. Extension of Theorem \ref{['Berry-Esseen Bounds for Statistics of Weakly Dependent Samples Proof']} to Polynomial Mixing Rates for Stationary Sequences
  8. Applications of Theorem \ref{['Berry-Esseen Bounds for Statistics of Weakly Dependent Samples Proof']}
  9. A General Berry–Esseen Theorem for Non-Stationary Sequences With Non-Uniform Third Moments
  10. Application to Uniformly Bounded Summands Without Assumptions on the Variance Growth Rate
  11. Alternative Bounds for Remainder Terms
  12. U-Statistics
  13. Functions of Sample Means
  14. Studentized Sample Means
  15. Conclusion and Future Avenues
  16. ...and 2 more sections

Key Result

Theorem 1

(Berry-Esseen Bounds for Statistics of $\phi$-Mixing, Non-Stationary Sequences, with Polynomial Rates and Linear Variance) Assume that the above four assumptions hold. For each $1\le j\leq k\leq N$, let denote a $\sigma^C[j,k]$-measurable random variable such that $R_{j,j-1} = R$ for all $2 \leq j\leq N$. For each $N\in\mathbb{Z}^{+}$, define, for any $\varepsilon >0$, Then, for a choice of $p$

Theorems & Definitions (18)

  • Theorem 1
  • Remark 1
  • Lemma 2: Decoupling Inequality
  • Lemma 3: Rosenthal's Inequality for $\phi$-mixing sequences
  • Lemma 4
  • Theorem 5: Berry-Esseen Bounds without Uniform Third Moment Bound
  • Remark 2
  • Corollary 5.1
  • Theorem 6: Alternative Remainder Bounds for Theorem \ref{['Non Uniform Third Moment Application']} for Statistics with Additional Remainder
  • Theorem 7: Berry-Esseen Bounds for U-Statistics
  • ...and 8 more