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It defines the independent, mean-zero triangular array setting, establishes the Lindeberg condition through the Lindeberg function, and proves convergence of distribution for normalized sums to the standard normal. The argument uses convergence of smooth test-function expectations, step-function approximation via sandwiching between smooth functions, and an approximation/bracketing strategy. It further derives Lindeberg–Levy and Lyapunov versions with their respective 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":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/central-limit-theorems-and-proofs-read-online-notes/268519/",{"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/central-limit-theorems-and-proofs-read-online-notes/268519.png","ImageObject",300,407,{"name":92,"@type":93},"Quinn Holloway","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-09-21","2026-09-15",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the core setting assumed for proving the central limit theorem here?","Question",{"text":112,"@type":113},"The document assumes independent random variables with mean 0 and variances that sum to a positive quantity 𝜎_n^2 for each n, defining S_n as their sum.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What role does the Lindeberg condition play in the theorem?",{"text":117,"@type":113},"The theorem states that if the Lindeberg function L_n(Ξ΅) tends to 0 for every Ξ΅>0, then the probability distribution of the normalized sum converges to the standard normal CDF.",{"name":119,"@type":110,"acceptedAnswer":120},"How are the Lindeberg–Levy and Liapunov CLTs obtained from the main Lindeberg result?",{"text":121,"@type":113},"The document explains that other special versions follow immediately: Lindeberg–Levy assumes i.i.d. variables with finite positive variance, while Liapunov assumes independent (not necessarily identical) variables with finite third absolute central moments, each providing a sufficient condition for asymptotic normality.","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},268519,1789971210,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":46},2336474466712,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Math/Stat 394, Winter 2019 \nF.W. Scholz \nCentral Limit Theorems and Proofs \nThe following gives a self-contained treatment of the central limit theorem (CLT) . It is based on Lindeberg's (1922) method. To state the CLT which we shall prove, we introduce the following notation. We assume that Xn1; : : : ; Xnn are independent random variables with means 0 and respective variances τ€€›2n1 ; : : : ; τ€€›2nn with \nτ€€›2n1 + : : : + τ€€›2nn = τ€€œ2n > 0 for all n \nDenote the sum Xn1 + : : : + Xnn by Sn and observe that Sn has mean zero and variance τ€€œ2n, see Fact 8 .28 and 8 .31 in Anderson et al. \nLindeberg's Central Limit Theorem: \nIf the Lindeberg condition is satisτ€€Œed, i.e. , if for every τ€€ > 0 we have that \n 1 Ln (τ€€) = τ€€œ2n \nn \nXE τ€€X2niIfjXnijτ€€•τ€€τ€€œngτ€€‘ τ€€€! 0 as n ! 1 ; i=1 \nthen for every a 2 R we have that \nP (Sn =τ€€œn τ€€” a) τ€€€ τ€€ˆ(a) τ€€€! 0 as n ! 1 \nProof: Step 1 (convergence of expectations of smooth functions): We will show in Appendix 1 that for certain functions f we have that \nE [f (Sn =τ€€œn )] τ€€€ E [f (Z)] ! 0 as n ! 1 ; (1) \nwhere Z denotes a standard normal random variable. If this convergence would hold for any function f and if we then applied it to \nfa (x) = I(τ€€€1 ;a](x) = 1 if x τ€€” a and = 0 if x > a; \nthen \nE [fa (Sn =τ€€œn )] τ€€€ E [fa (Z)] = P (Sn =τ€€œn τ€€” a) τ€€€ τ€€ˆ(a) \nand the statement of the CLT would follow. Unfortunately, we cannot directly demonstrate the above convergence (1) for all f , but only for smooth f. Here smooth f means that f is bounded and has three bounded, continuous derivatives as stipulated in Lemma 1 of Appendix 1 . \nStep 2 (sandwiching a step function between smooth functions): We will approximate fa (x) = I(τ€€€1 ;a](x), which is a step function with step at x = a, by sandwiching it between two smooth functions. In fact, for τ€€Ž > 0 one easily τ€€Œnds (see Appendix 2 for an explicit example) smooth functions f (x) with f (x) = 1 for x τ€€” a, f (x) monotone decreasing from 1 to 0 on [a; a + τ€€Ž] and f (x) = 0 for x τ€€• a + τ€€Ž . Hence we would have \nfa (x) τ€€” f (x) τ€€” fa+τ€€Ž(x) for all x 2 R \n\n| fa(x) | \u003Cbr>f(x) | fa+Ξ΄(x) |\n| --- | --- | --- |\n\na a + Ξ΄ \nSince fa+τ€€Ž(x + τ€€Ž) = fa (x), we also get from the second previous \\τ€€” \" \nfa (x) = fa+τ€€Ž (x + τ€€Ž) τ€€• f (x + τ€€Ž) \nCombining these, we can bracket fa (x) for all x 2 R by \nf (x + τ€€Ž) τ€€” fa (x) τ€€” f (x) \n\n| f(x + Ξ΄) | fa(x) | \u003Cbr>f(x) |\n| --- | --- | --- |\n\na βˆ’ Ξ΄ a a + Ξ΄ \nStep 3 (the approximation argument): The following diagram will clarify the approximation strategy. The inequalities result from the bracketing in the previous step. \nE hf τ€€ Sτ€€œ~~n~~n + τ€€Ž τ€€‘i τ€€” E hfa τ€€ Sτ€€œ~~n~~n τ€€‘i τ€€” E hf τ€€ Sτ€€œ~~n~~n τ€€‘i \noo oo \nE [f (Z + τ€€Ž)] τ€€” E [fa (Z)] τ€€” E [f (Z)] \noo means that for any τ€€Œxed τ€€Ž > 0 and for τ€€Œxed f the terms above and below oo become arbitrarily close, say within τ€€=3 of each other, as n ! 1 (see Step 1) . Since f (Z + τ€€Ž) τ€€€ f (Z) = 0 for \nZ 26 [a τ€€€ τ€€Ž; a + τ€€Ž] and j f(Z + τ€€Ž) τ€€€ f(Z)j τ€€” 1 for a τ€€€ τ€€Ž τ€€” Z τ€€” a + τ€€Ž we have 1 \njE [f (Z + τ€€Ž)] τ€€€ E [f (Z)]j = τ€€Œ E h (f (Z + τ€€Ž) τ€€€ f (Z)) I[aτ€€€τ€€Žτ€€”Zτ€€”a+τ€€Ž]i τ€€Œ τ€€” E τ€€Œ hI[aτ€€€τ€€Žτ€€”Zτ€€”a+τ€€Ž]i τ€€Œ \n= P (a τ€€€ τ€€Ž τ€€” Z τ€€” a + τ€€Ž) = τ€€ˆ(a + τ€€Ž) τ€€€ τ€€ˆ(a τ€€€ τ€€Ž) \nτ€€” τ€€ˆ(τ€€Ž) τ€€€ τ€€ˆ(τ€€€τ€€Ž) τ€€” 2τ€€Žτ€€ž(0) = 2τ€€Ž= p2τ€€™ \nThe latter bound can be made as small as τ€€=3, no matter what f is, by taking τ€€Ž = τ€€ p2τ€€™=6. With this choice of τ€€Ž, and n suτ€€Žciently large, the above diagram entails that \nτ€€Œ E τ€€” fa τ€€’ Sτ€€œnn τ€€“τ€€• τ€€€ E [fa (Z)] τ€€Œ τ€€” τ€€3 + τ€€3 + τ€€3 = τ€€ Since this can be done for any τ€€ > 0 we have shown that \nτ€€Œ E τ€€” fa τ€€’ Sτ€€œnn τ€€“τ€€• τ€€€ E [fa (Z)] τ€€Œ ! 0 as n ! 1 2 \nFrom Lindeberg's CLT other special versions follow at once. The τ€€Œrst, the Lindeberg-Levy CLT, considers independent identically distributed random variables with τ€€Œnite variance τ€€› 2 . The second, the Liapunov CLT, considers independent, but not necessarily identically distributed random variables with τ€€Œnite third moments. \nLindeberg-Levy CLT: Let Y1 ; : : : ; Yn be independent and identically distributed random vari \nables with common mean τ€€– and τ€€Œnite positive variance τ€€› 2 and let Tn = Y1 + : : : + Yn. Then for all a 2 R P ~~ ~~Tnpnτ€€›τ€€– τ€€” a! τ€€€ τ€€ˆ(a) τ€€€! 0 as n ! 1 \nProo","cbCaiiPm9gW24Ne4","https://ap.wps.com/l/cbCaiiPm9gW24Ne4","pdf",342372,"English","# Central Limit Theorem and Notation\n## Lindeberg's Central Limit Theorem\n## Proof Structure: Step 1-3\n# Derived Special Cases\n## Lindeberg–Levy CLT\n## Liapunov CLT\n# Appendix 1: Taylor Formula","[{\"question\":\"What is the core setting assumed for proving the central limit theorem here?\",\"answer\":\"The document assumes independent random variables with mean 0 and variances that sum to a positive quantity 𝜎_n^2 for each n, defining S_n as their sum.\"},{\"question\":\"What role does the Lindeberg condition play in the theorem?\",\"answer\":\"The theorem states that if the Lindeberg function L_n(Ξ΅) tends to 0 for every Ξ΅\\u003e0, then the probability distribution of the normalized sum converges to the standard normal CDF.\"},{\"question\":\"How are the Lindeberg–Levy and Liapunov CLTs obtained from the main Lindeberg result?\",\"answer\":\"The document explains that other special versions follow immediately: Lindeberg–Levy assumes i.i.d. variables with finite positive variance, while Liapunov assumes independent (not necessarily identical) variables with finite third absolute central moments, each providing a sufficient condition for asymptotic normality.\"}]","Central Limit Theorems and Proofs - read online notes | PDF",1789431966]