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The proof relies on systematic combinatorial enumeration, generating-function analysis, and concentration inequalities of the Poincar type. Results include generalized Wigner and Wishart matrices, with explicit limiting variance formulas and extensions to polynomial and continuously differentiable test functions under stated moment conditions.",{"@graph":69,"@context":122},[70,84,105],{"@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/a-clt-for-a-band-matrix-model-june-15-2004/268671/",{"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/a-clt-for-a-band-matrix-model-june-15-2004/268671.png","ImageObject",300,407,{"name":92,"@type":93},"Bill Black","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-09","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":34},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What type of random matrices are studied in this work?","Question",{"text":112,"@type":113},"The paper considers random symmetric matrices whose on-or-above diagonal entries are independent, not necessarily identically distributed and not necessarily sharing the same variance.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What are the main probabilistic results derived?",{"text":117,"@type":113},"A law of large numbers and a central limit theorem are proved for linear statistics of the empirical eigenvalue distribution.",{"name":119,"@type":110,"acceptedAnswer":120},"How is the central limit theorem obtained from polynomial to smooth test functions?",{"text":121,"@type":113},"The approach combines a combinatorial enumeration scheme with concentration inequalities of the Poincar type to extend beyond polynomial test functions.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},268671,1789434504,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":34,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},24189269381491,"https://ap-avatar.wpscdn.com/avatar/160000cf11732dd8392?x-image-process=image/resize,m_fixed,w_180,h_180&k=1788146458752108895","A CLT FOR A BAND MATRIX MODEL  \nGREG ANDERSON AND OFER ZEITOUNI  \nAbstract. A law of large numbers and a central limit theorem are derived for linear statistics of random symmetric matrices whose on-or-above diagonal entries are independent, but neither necessarily identically distributed, nor necessarily all of the same variance. The derivation is based on systematic combinatorial enumeration, study of generating functions, and concentration inequalities of the Poincar type. Special cases treated, with an explicit eval  \nuation of limiting variances, are generalized Wigner and Wishart matrices.  \n1. Introduction  \nThe interest in the limiting properties of the empirical distribution of eigenvalues of large symmetric random matrices can be traced back to [Wis28] and to the pathbreaking article of Wigner [Wig55] . We refer to [Ba99], [De00],[PH00],[Me91] and [PL03] for partial overview and some of the recent spectacular progress in this 􀀌eld. In this paper we study both convergence of the empirical distribution and central limit theorems for linear statistics of the empirical distribution of a class of random matrices. To give right away the 􀀍avor of our results, consider for each positive integer N the N-by-N symmetric random matrix X(N ) with on-or-above-diagonal entries X (N )ij = N 􀀀1=2f (i=N; j=N )1=2􀀘ij , where the 􀀘ij are zero mean unit variance i.i.d. random variables satisfying the Poincar inequality with constant c (see x11.4 for the de􀀌nition), and f (􀀁 ; 􀀁) is a nonnegative function symmetric and continuous on [0 ; 1]2 such that R01 f (x; y)dy 􀀑 1. De􀀌ne the semicircle distribution 􀀛Sof zero mean and unit variance to be the measure on R of compact support with  \ndensity ~~d􀀛~~dSx := 2~~1~~􀀙 p4 􀀀 x21fj xj􀀔2g . Let 􀀕 1 (N ) 􀀔 􀀁 􀀁 􀀁 􀀔 􀀕 N (N ) be the eigenvalues of X(N ) . Under these assumptions, a corollary of our general results (see Theorem 3.5 below) states that the empirical distribution L (N ) := N 􀀀1 P 􀀎􀀕 i (N) converges weakly, in probability, to 􀀛S , and further, for any continuously di􀀋erentiable function f on R of polynomial growth, with kf0 kL2 (􀀛S ) > 0, the sequence of random variables  \n(1) Xi1 f (􀀕i (N )) 􀀀 E \"Xi1 f (􀀕i (N ))\\#  \nconverges in distribution to a nondegenerate zero mean Gaussian random variable with variance given by an explicit formula. Similar explicit results hold for a class of generalized Wishart matrices, see Theorem 12.8. For polynomial test functions f , such results hold in much greater generality, see Theorem 3.3.  \nOur approach has two main components. The 􀀌rst, of some interest on its own, is a combinatorial enumeration scheme for the di􀀋erent types of terms that contribute  \nDate: June 15, 2004.  \nO.Z. was partially supported by NSF grant number DMS-0302230 .  \n2 GREG ANDERSON AND OFER ZEITOUNI  \nto the expectation of products of traces of powers of the matrices under study. This scheme takes the bulk of the paper to develop. The other component, which allows us to move from polynomial test functions to continuously di􀀋erentiable ones, is based on concentration inequalities of the Poincar type. The latter component is developed in x11, building on earlier results of concentration for random matrices that can be found in [CB04] and [GZ00] .  \nCLT results related to Theorems 3.3 and 3.5 below have already been stated in the literature. An especially strong inspiration to our study is the work of Jonsson [Jo82], who gives CLT statements for traces of polynomial functions of Gaussian Wishart matrices, based on the method of moments introduced by Wigner in [Wig55] . The method of moments was revisited in the far-reaching work of Sinai and Soshnikov [SS98], where, as an easy by-product of their results, they state a CLT for traces of analytic functions of Wigner-type matrices. Pastur and co-authors, on the one hand, and Bai and co-authors, on the other, have championed an approach based on the evaluation of resolvents. The latter approach has the advantage of allowing one to relax ","cbCaifkjhQZgtVg6","https://ap.wps.com/l/cbCaifkjhQZgtVg6","pdf",369569,46,"English","# Introduction\n## Linear eigenvalue statistics and semicircle limit\n## Combinatorial enumeration and generating functions\n## Concentration inequalities and extensions to test functions","[{\"question\":\"What type of random matrices are studied in this work?\",\"answer\":\"The paper considers random symmetric matrices whose on-or-above diagonal entries are independent, not necessarily identically distributed and not necessarily sharing the same variance.\"},{\"question\":\"What are the main probabilistic results derived?\",\"answer\":\"A law of large numbers and a central limit theorem are proved for linear statistics of the empirical eigenvalue distribution.\"},{\"question\":\"How is the central limit theorem obtained from polynomial to smooth test functions?\",\"answer\":\"The approach combines a combinatorial enumeration scheme with concentration inequalities of the Poincar type to extend beyond polynomial test functions.\"}]","A CLT FOR A BAND MATRIX MODEL - June 15, 2004 | PDF",116]