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Berry-Esseen Theorem for Sample Quantiles with Locally Dependent Data

Partha S. Dey, Grigory Terlov

TL;DR

This work derives a Gaussian Central Limit Theorem for sample quantiles under locally dependent data, providing an explicit convergence rate through a reduction to sums of indicator variables and the use of Stein's method for local dependence. It establishes univariate and multivariate CLTs with concrete finite-sample bounds that depend on dependency-neighborhood sizes $D_1,D_2,D_3$, and extends naturally to joint convergence of multiple quantiles with a covariance structure $\Sigma_n$. An application to moving-average models shows Gaussian approximations hold when the MA order $q$ satisfies $q\ll n^{1/4}$, and the paper discusses potential rate-optimality, including an i.i.d. case where the rate is $O(\log n/\sqrt{n})$ and a conjectured $O(n^{-1/2})$-type refinement. Overall, the results provide practical Gaussian approximations for quantile-based statistics under broad local-dependence structures, enabling precise inference for dependent time-series quantiles. $

Abstract

We derive a Gaussian Central Limit Theorem for the sample quantiles based on locally dependent random variables with explicit convergence rate. Our approach is based on converting the problem to a sum of indicator random variables, applying Stein's method for local dependence, and bounding the distance between two normal distributions. We also generalize this approach to the joint convergence of sample quantiles with an explicit convergence rate.

Berry-Esseen Theorem for Sample Quantiles with Locally Dependent Data

TL;DR

This work derives a Gaussian Central Limit Theorem for sample quantiles under locally dependent data, providing an explicit convergence rate through a reduction to sums of indicator variables and the use of Stein's method for local dependence. It establishes univariate and multivariate CLTs with concrete finite-sample bounds that depend on dependency-neighborhood sizes , and extends naturally to joint convergence of multiple quantiles with a covariance structure . An application to moving-average models shows Gaussian approximations hold when the MA order satisfies , and the paper discusses potential rate-optimality, including an i.i.d. case where the rate is and a conjectured -type refinement. Overall, the results provide practical Gaussian approximations for quantile-based statistics under broad local-dependence structures, enabling precise inference for dependent time-series quantiles. $

Abstract

We derive a Gaussian Central Limit Theorem for the sample quantiles based on locally dependent random variables with explicit convergence rate. Our approach is based on converting the problem to a sum of indicator random variables, applying Stein's method for local dependence, and bounding the distance between two normal distributions. We also generalize this approach to the joint convergence of sample quantiles with an explicit convergence rate.
Paper Structure (15 sections, 9 theorems, 105 equations)

This paper contains 15 sections, 9 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Stein's method
  3. Notations
  4. Setup
  5. Main results
  6. Organization
  7. Application to the moving-average model
  8. Preliminary Results
  9. Local dependence
  10. Bound on the distance between two Gaussian vectors with the same mean
  11. Variance Control
  12. Proof of Main Results
  13. Limiting distribution of the sample median
  14. Limiting joint distribution of sample quantiles
  15. Optimal rate of convergence

Key Result

Theorem 1.2

Let $(X_i)_{i\in[n]}$ be a sequence of random variables satisfying Assumption as1 with $\alpha=1/2$, $m_\alpha=0$ and Assumption as:dn. Let $M_n$ be as in def:M, $A$ be as in eq:unifbdd, $\theta_n$ be as in def:cmed, $D_i$ as in eq:D, $\sigma_n$ as in def:sigmas, and $Z\sim\mathop{\mathrm{N}}\nolimi Then

Theorems & Definitions (24)

  • Definition 1.1: Dependency graph
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 3.1: Theorem 2.1 and Remark 2.2 in Fang16
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 14 more