Newman's CLT via Stein's method - A new proof under additional assumptions

Q: What does Newman’s CLT address in this work?

It addresses a Central Limit Theorem for strictly stationary sequences that are linearly positive or linearly negative quadrant dependent, provided the covariance series converges under a finite second-moment condition.

Q: How does this paper obtain a new proof of Newman’s CLT?

It uses Stein’s method to reprove Newman’s CLT, adding assumptions of positive association and finite third-moment to fit the standard requirements for Stein-equation bounds.

Q: What role do quadrant dependence and association play?

The paper defines positive/negative quadrant dependence and then introduces stronger dependence notions such as positive/negative association. These dependence structures are the foundation for applying the CLT and for making Stein’s method estimates work under the stated moment conditions.

Sep 15, 2026

Uploaded by Angel

In 1980, C. M. Newman proved the Central Limit Theorem for strictly stationary sequences that are linearly positive or linearly negative quadrant dependent, under a finite second-moment condition ensuring convergence of the covariance series. This paper presents a new proof of Newman’s CLT using Stein’s method, introducing stronger assumptions. The approach relies on positive association and finite third-moment requirements, matching the usual needs of Stein-type arguments.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages13

Reading Time33 min

AccessFree to download

Abstract

Introduction

Quadrant dependence and related concepts

Strict stationarity and Newman’s CLT

Stein's method and the Stein operator

Key lemmas (Stein equation, bounds)

Main results (CLT via Stein’s method)

Proving Newman's CLT via Stein's Method Bruno Barbosa􀀃, Wasamon Jantai􀀃, Yevgeniy Kovchegov􀀃, and Guochen Xu􀀃 Abstract In 1980 C. M. Newman has proven the Central Limit Theorem for a sequence of strictly stationary random variables fXng n2N that are linearly positive or negative 1 quadrant dependent with convergent P Cov(X1 ; Xi) . In this paper, we present a new i=2 proof of the Newman's CLT based on Stein's method under additional assumptions. 1 Introduction In the past century, a lot of the e􀀋ort had been dedicated to proving the Central Limit Theorem (CLT) under a variety of dependence restrictions. C. M. Newman [18, 19, 20] has proven the CLT for the sequences of strictly stationary linearly positive or linearly negative quadrant dependent random variables under 􀀌nite second moment assumption. In this paper, we will revisit the Newman's CLT, and reprove it using the Stein's method under additional association and 􀀌nite third moment assumptions. Note that the 􀀌nite third moment assumption is usual for the Stein's method. In 1966, Lehmann [14] introduced the notion of positive and negative quadrant dependencies. De􀀌nition 1 . A pair of random variables, X and Y , is positive quadrant dependent (PQD) if P [X > x; Y > y] 􀀀 P[X > x]P [Y > y] 􀀕 0 ; 8x; y 2 R: (1) Consequently, a pair of random variables, X and Y , is said to be negative quadrant dependent (NQD) if X and 􀀀Y are positive quadrant dependent. There are stronger dependencies de􀀌ned as follows. De􀀌nition 2 . A sequence of random variables Xn is linearly positive quadrant dependent (LPQD) if for any pair of disjoint sets of indices A and B, and a positive sequence f􀀕ig, the random variables 􀀆 i2A􀀕iXi and 􀀆j2B􀀕j Xj are positive quadrant dependent. Similarly, Xn is linearly negative quadrant dependent (LNQD) if for any pair of disjoint sets of indices A and B, and a positive sequence f􀀕ig, the random variables 􀀆i2A􀀕i Xi and 􀀆j2B􀀕j Xj are negative quadrant dependent. The above de􀀌ned LPQD and LNQD are weaker types of dependencies than the positive and negative associations that were originally introduced by Esary at al. in [10] . 􀀃 Department of Mathematics, Oregon State University, Corvallis, OR 97331 De􀀌nition 3 . A 􀀌nite set of random variables fX1 ; X2 ; : : : ; Xn g is said to be positively associated (or simply, associated) if for any pair A and B of subsets of f1 ; 2; : : : ; ng the following holds Cov 􀀀f(Xi ; i 2 A); g (Xj ; j 2 B)􀀁 􀀕 0 for all coordinate-wise increasing functions f on RjAj and f on Rj Bj , whenever the respective covariances exist. A 􀀌nite set of random variables fX1 ; X2 ; : : : ; Xn g is said to be negatively associated if for any pair A and B of subsets of f1 ; 2; : : : ; ng the following holds Cov 􀀀f(Xi ; i 2 A); g (Xj ; j 2 B)􀀁 􀀔 0 for all coordinate-wise increasing functions f on RjAj and f on Rj Bj , whenever the respective covariances exist. An in􀀌nite sequence of random variables fXngn2N is said to be positively associated or associated (respectively, negatively associated) if every 􀀌nite subfamily is positively associated (respectively, negatively associated) . Next, we recall the de􀀌nition of strict stationarity. De􀀌nition 4 . A sequence fXngn2N is said to be strictly stationary if for all k; n 2 N, and all indices t 1 < t2 < 􀀁 􀀁 􀀁 < tn in N, the random vector (Xt 1+k; Xt 2+k; : : : ; Xtn+k) has the same joint distribution as (Xt 1; Xt 2; : : : ; Xtn) . In 1980, Newman [18] established the following version of CLT for positively associated strictly stationary sequences. Theorem 1 (Newman's CLT,[18, 21]) . Consider an LPQD strictly stationary sequence 1 fXngn2N 2 L2 such that the series P Cov(X1 ; Xj) converges. Then, j=2 􀀛~~ ~~1p~~n~~ 􀀀 Xj 􀀀 E[Xj]􀀁 􀀀d! Z; (2) where 1 􀀛 2 = Var(X1 ) + 2 XCov (X1 ; Xj) ; j=2 Z denotes the standard normal random variable, and 􀀀d! in (2) refers to convergence in distribution. Moreover, Newman [18] proved a corresponding version o

FAQs

What does Newman’s CLT address in this work?

It addresses a Central Limit Theorem for strictly stationary sequences that are linearly positive or linearly negative quadrant dependent, provided the covariance series converges under a finite second-moment condition.

How does this paper obtain a new proof of Newman’s CLT?

It uses Stein’s method to reprove Newman’s CLT, adding assumptions of positive association and finite third-moment to fit the standard requirements for Stein-equation bounds.

What role do quadrant dependence and association play?

The paper defines positive/negative quadrant dependence and then introduces stronger dependence notions such as positive/negative association. These dependence structures are the foundation for applying the CLT and for making Stein’s method estimates work under the stated moment conditions.

Sep 15, 2026

Uploaded by Angel