[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-268531-105":59,"doc-detail-268531-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","newmans-clt-via-steins-method-a-new-proof-under-additional-assumptions","Newman's CLT via Stein's method - A new proof under additional assumptions","","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.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/newmans-clt-via-steins-method-a-new-proof-under-additional-assumptions/268531/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/newmans-clt-via-steins-method-a-new-proof-under-additional-assumptions/268531.png","ImageObject",300,407,{"name":92,"@type":93},"Angel","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-01","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does Newman’s CLT address in this work?","Question",{"text":113,"@type":114},"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.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does this paper obtain a new proof of Newman’s CLT?",{"text":118,"@type":114},"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.",{"name":120,"@type":111,"acceptedAnswer":121},"What role do quadrant dependence and association play?",{"text":122,"@type":114},"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.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},268531,1789432049,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},687207412472,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","Proving Newman's CLT via Stein's Method Bruno Barbosa􀀃, Wasamon Jantai􀀃, Yevgeniy Kovchegov􀀃, and Guochen Xu􀀃  \nAbstract  \nIn 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  \n1  \nquadrant dependent with convergent P Cov(X1 ; Xi) . In this paper, we present a new  \ni=2  \nproof of the Newman's CLT based on Stein's method under additional assumptions.  \n1 Introduction  \nIn 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.  \nIn 1966, Lehmann [14] introduced the notion of positive and negative quadrant dependencies.  \nDe􀀌nition 1 . A pair of random variables, X and Y , is positive quadrant dependent (PQD) if  \nP [X > x; Y > y] 􀀀 P[X > x]P [Y > y] 􀀕 0 ; 8x; y 2 R: (1)  \nConsequently, a pair of random variables, X and Y , is said to be negative quadrant dependent (NQD) if X and 􀀀Y are positive quadrant dependent.  \nThere are stronger dependencies de􀀌ned as follows.  \nDe􀀌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.  \nSimilarly, 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.  \nThe 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] .  \n􀀃 Department of Mathematics, Oregon State University, Corvallis, OR 97331  \nDe􀀌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  \nCov 􀀀f(Xi ; i 2 A); g (Xj ; j 2 B)􀀁 􀀕 0  \nfor all coordinate-wise increasing functions f on RjAj and f on Rj Bj , whenever the respective covariances exist.  \nA 􀀌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  \nCov 􀀀f(Xi ; i 2 A); g (Xj ; j 2 B)􀀁 􀀔 0  \nfor all coordinate-wise increasing functions f on RjAj and f on Rj Bj , whenever the respective covariances exist.  \nAn 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) .  \nNext, we recall the de􀀌nition of strict stationarity.  \nDe􀀌nition 4 . A sequence fXngn2N is said to be strictly stationary if for all k; n 2 N, and all indices t 1 \u003C t2 \u003C 􀀁 􀀁 􀀁 \u003C 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) .  \nIn 1980, Newman [18] established the following version of CLT for positively associated strictly stationary sequences.  \nTheorem 1 (Newman's CLT,[18, 21]) . Consider an LPQD strictly stationary sequence  \n1  \nfXngn2N 2 L2 such that the series P Cov(X1 ; Xj) converges. Then,  \nj=2  \n􀀛~~ ~~1p~~n~~  􀀀 Xj 􀀀 E[Xj]􀀁 􀀀d! Z; (2)  \nwhere  \n1  \n􀀛 2 = Var(X1 ) + 2 XCov (X1 ; Xj) ;  \nj=2  \nZ denotes the standard normal random variable, and 􀀀d! in (2) refers to convergence in distribution.  \nMoreover, Newman [18] proved a corresponding version o","cbCaioKeJisAnq7b","https://ap.wps.com/l/cbCaioKeJisAnq7b","pdf",274595,13,"English","# Abstract\n# Introduction\n## Quadrant dependence and related concepts\n## Strict stationarity and Newman’s CLT\n## Stein's method and the Stein operator\n## Key lemmas (Stein equation, bounds)\n# Main results (CLT via Stein’s method)","[{\"question\":\"What does Newman’s CLT address in this work?\",\"answer\":\"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.\"},{\"question\":\"How does this paper obtain a new proof of Newman’s CLT?\",\"answer\":\"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.\"},{\"question\":\"What role do quadrant dependence and association play?\",\"answer\":\"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.\"}]","Newman's CLT via Stein's method - A new proof under additional assumptions | PDF",33]