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It assumes short-range dependence and measurable G with finite expectation of |G(X1)|, then derives weak convergence of partial-sum processes to a scaled Brownian motion in the Skorokhod function space. The proof uses martingale decomposition of Yn=G(Xn)-EG(Xn) and applies auxiliary martingale-difference CLTs and weak-convergence results.",{"@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/some-remarks-on-the-central-limit-theorem-for-functionals-of-linear-processes-under-short-range-dependence/268577/",{"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/some-remarks-on-the-central-limit-theorem-for-functionals-of-linear-processes-under-short-range-dependence/268577.png","ImageObject",300,407,{"name":92,"@type":93},"Kurz","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",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What type of time series models are considered in this paper?","Question",{"text":113,"@type":114},"The paper considers one-sided m-dimensional linear processes, and also introduces a related multidimensional linear process for the innovations vector.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What is the main probabilistic limit proved for functionals of the linear processes?",{"text":118,"@type":114},"Under short-range dependence and Lipschitz-type regularity conditions on G, the normalized partial-sum process converges in distribution to σ times a standard Brownian motion on [0,1].",{"name":120,"@type":111,"acceptedAnswer":121},"How does the proof establish the central limit theorem?",{"text":122,"@type":114},"It uses a martingale decomposition of Yn=G(Xn)-E[G(Xn)], then combines martingale difference CLT results with weak convergence lemmas in the Skorokhod space.","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},268577,1789432536,{"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":105,"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":130,"read_time":144,"preview_page_count":4},2336478945635,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","# SOME REMARKS ON THE CENTRAL LIMIT THEOREMFOR FUNCTIONALS OF LINEAR PROCESSESUNDER SHORT-RANGE DEPENDENCE\n\nBY  \nKONRAD FURMANCZYK(WARSZAWA)  \nAbstract.In this paper we consider the central limit theorems forfunctionals G:R\"→R of one-sided m-dimensional linear processes,where A,is a nonrandom matrix m×mand Z's are  \ni.i.d.random vectors in R\".  \n2000 AMS Mathematics Subject Classification:Primary:60F05;  \nSecondary:60G10.  \nKey words and phrases:Central limit theorem,linear process,time series,short-range dependence,Markov chain,martingale diffe-rences.  \n## 1.INTRODUCTION\n\nThe one-sided linear process(LP)is defined as follows:  \nj=1,2,…,n,  \nwhere the innovationsare i.i.d.random variables and ai are constantcoefficients such that  \nLinear processes have a wide range of applications in time series analysis.A large class of time series processes can be modelled in such a way,includinga subset of the fractional ARIMA processes (see Brockwell and Davis(1987).Additionally,we say that(Xn)is a short-range dependent process if the followingcondition is satisfied:  \nIt is obvious that ifand E(Z)\u003C∞,then the linearprocess(Xn)is short-range dependent.  \nK.Furmańczyk  \n   \nLet G be a measurable function such that E|G(X₁)|\u003C∞.In statisticalinference for time series,it is an important problem to investigate the asymp-totic behavior of the partial sums,since many statis-tics are of this form.This problem has been considered by many authors.Forexample,Davydov(1970)considers the special case G(x)=x,which is alsodiscussed in Giraitis and Surgailis (1986).This problem has been consideredrecently by Ho and Hsing (1996),(1997),Hsing (2000),Koul and Surgailis(2001),Wu (2002).  \nThe central limit theorem for multiple linear processes with heavy tailedinnovations has been proved by Wu(2003).A very simple question is whetherone can prove a central limit theorem (CLT)under a short-range dependencewithout additional assumptions on the (a,)and (Zn)and for the most generalclass of functions G.  \nThe generalization of the one-sided linear process is the multidimensionallinear process(MLP),defined as follows:  \nare i.i.d.random vectors in Rmwhere the innovationswith mean zero and  \nE|Z₁I²=1.  \n(1.1)  \n,Ao=I,and theThe(A)are the nonrandom matrices,wherematrix norm ||ll is such that,for every z∈R\",  \n(1.2)  \n|A,z|≤||AIIIz|,  \nwhere ||is the usual Euclidean norm.We will also consider the following assump-tions:  \n(ao)  \nfor some t>1,  \n(a₂)  \n(b₁)the density of the vector Z₁is the Lipschitz function,  \n(b₂(t))E|Z₁I\u003C∞  for some t≥2.  \nIn Section 4 we prove the CLT for the sums of functionals of linearprocesses.Let ⑨[0,1]be the space of all real functions defined on [0,1]thatare right-continuous and have left-side limits with Skorokhod topology (seeBillingsley(1968)).We will show that,under some simple conditions,  \nn-1/2Sm→σ{B(t),0≤t≤1}  in ⑨[0,1]for some σ.  \n   \nThe basic idea of this proof is the martingale decomposition of the process  \nYn=G(Xn)-EG(Xn).  \nThis decomposition is of the following form (see also Ho and Hsing(1996)and (1997)):  \nwhere Un,s=E{Yn|n-s-E{Yn|9n-s-1},(1.3)  \nwhere n:=σ(…,Zn-1,Zn)is the σ-field generated by the innovations in the“past”≤n,(Xn)is an m-dimensional process(MLP),and G:Rm→R is a realfunction such that E|Y₁I⁰\u003C∞for some Q≥2.  \nIn the proof of the main theorem we use the idea introduced by Koul andSurgailis(1997).  \nTheorem 2.2 is a simple corollary to the theorem of Maxwell and Wood-roofe (see Wu(2002),Corollary 1).In this context see also Ho and Hsing(1997).They proved that ifand(ao)holds,then under someregularity conditions on G (for example,if the derivative of G is bounded andcontinuous,or G is any polynomial)  \nSn/√n→N(0,o²)  for some σ\u003C0  \n(see Theorem 4.1 in Ho and Hsing (1997).  \nOn the other hand,under the assumptions that  \nis Lipschitz for all sufficiently large n,an=n⁻L(n)for some y>1,where L()is a slowly varying function,Wu(2002)proved the CLT.  \n## 2.MAIN RESULTS\n\nThe main results are the central limit theorems fo","cbCaib6daPk9Fa0t","https://ap.wps.com/l/cbCaib6daPk9Fa0t","pdf",445930,"English","# 1. INTRODUCTION\n# 2. MAIN RESULTS\n# 3. AUXILIARY LEMMAS","[{\"question\":\"What type of time series models are considered in this paper?\",\"answer\":\"The paper considers one-sided m-dimensional linear processes, and also introduces a related multidimensional linear process for the innovations vector.\"},{\"question\":\"What is the main probabilistic limit proved for functionals of the linear processes?\",\"answer\":\"Under short-range dependence and Lipschitz-type regularity conditions on G, the normalized partial-sum process converges in distribution to σ times a standard Brownian motion on [0,1].\"},{\"question\":\"How does the proof establish the central limit theorem?\",\"answer\":\"It uses a martingale decomposition of Yn=G(Xn)-E[G(Xn)], then combines martingale difference CLT results with weak convergence lemmas in the Skorokhod space.\"}]","SOME REMARKS ON THE CENTRAL LIMIT THEOREMFOR FUNCTIONALS OF LINEAR PROCESSES UNDER SHORT-RANGE DEPENDENCE | PDF",28]