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Talk:Plurality voting

Isn't "two-round voting" much more common?
E.g., mayoral elections in Germany use it ... --User:Haraldmmueller 10:16, 18 September 2020 (UTC)

 * I don't think either one is much more common than the other. –Closed Limelike Curves (talk) 00:58, 22 February 2024 (UTC)

No mentioning of "election inversion" - why?
Similar question to Talk:First-past-the-post voting: Election inversion is a standard term; and a phenomenon mainly occurring with FPTP (albeit not only: Rounding procedures can produce them also in proportional systems). Why isn't this even mentioned here, let alone discussed - see of course 2000 and 2016 in the US (Michael Geruso, Dean Spears, Ishaana Talesara. 2019. "Inversions in US Presidential Elections: 1836-2016." NBER paper, slides by Nicholas R. Miller). It would nicely fit into the "wasted votes" section, because different amounts of "waste" for each selection directly imply the possibility of election inversion. --User:Haraldmmueller 08:20, 18 September 2020 (UTC)

 * It's not included because nobody's added it. Feel free to add it! Closed Limelike Curves (talk) 00:57, 22 February 2024 (UTC)

Similar articles
Plurality (voting) and this article seem to have same topic. Suggest merge. HudecEmil (talk) 19:36, 12 October 2022 (UTC)

Merge proposal
I am proposing Plurality (voting) be merged into Plurality voting. The articles have overlapping information and are about the same subject. Plurality voting is the longer article, is slightly higher quality, and receives more monthly pageviews (Plurality (voting) versus Plurality voting). --  Primium  (talk) 17:45, 13 January 2024 (UTC)

 * Previously brought this up. --  Primium  (talk) 17:50, 13 January 2024 (UTC)
 * @Primium the two topics are different, but the article you've linked is badly-named; I think we definitely don't want an article called "Plurality (voting)" that's different from "Plurality voting".
 * Plurality just means "the biggest share" (in British English, this is called a relative majority). It contrasts with a majority (in British English, an absolute majority), whch is a fraction bigger than 50%. Plurality (voting) could either be kept, turned into a Wiktionary entry, or merged into the majority page.
 * On the other hand, I just found out the page on first-past-the-post voting exists. That page is 100% a duplicate, since these are two names for the exact same system, and they need a merge. Closed Limelike Curves (talk) 02:32, 7 February 2024 (UTC)
 * Happy with a merge, but I think it should be to First-past-the-post voting (or simply First-past-the-post), which is the WP:COMMONNAME. Number   5  7  16:11, 24 February 2024 (UTC)
 * Google n-grams seems to suggest plurality is slightly more popular but I have no strong opinion on this. Closed Limelike Curves (talk) 03:15, 12 March 2024 (UTC)

 * Technically, there is a slight difference; first-past-the-post is the single-member version of plurality voting, which also includes multi-member versions. So merging into First-past-the-post doesn't make sense. Either merge into Plurality voting or shrink Plurality voting down to an overview of the different types of plurality voting, with a referral to First-past-the-post for a more detailed article on single-member plurality.174.67.226.163 (talk) 22:53, 9 March 2024 (UTC)
 * Oppose
 * Just, I oppose it. It's just obvious why
 * Cheers, Arotparaarms (talk) 15:53, 28 February 2024 (UTC)
 * Oppose for reasons stated above: it is a distinct concept, and in particularly common use among political-science discussion in the USA (it's far more common of a term). A merge would add confusion, not remove it. Denzera (talk) 14:51, 12 March 2024 (UTC)
 * Oh, I also oppose it but I didn't have a concrete reason, so I guess I now have a solid reason, and it's interesting how it's a common term across the globe. Arotparaarms (talk) 15:38, 12 March 2024 (UTC)
 * Oops, the wrong section. Thank you. Will move. Викидим (talk) 17:31, 13 April 2024 (UTC)
 * Oops, the wrong section. Thank you. Will move. Викидим (talk) 17:31, 13 April 2024 (UTC)

Merge from FPTP
FPTP and Plurality are the same system. FPTP should be merged into this article with a redirect. Closed Limelike Curves (talk) 17:21, 12 March 2024 (UTC)

 * Oppose FPTP is a particular case of plurality (and the most popular). What we need is to rewrite the lead to reflect this fact. --Викидим (talk) 06:18, 13 April 2024 (UTC)
 * This is asking about whether plurality (voting) should be merged with Plurality voting (which are two completely different things).
 * FPTP and plurality are synonyms.
 * Closed Limelike Curves (talk) 17:23, 13 April 2024 (UTC)
 * Many authors would disagree with your #2. Plurality voting, in many sources at least, includes such non-FPTP systems as two-round system and alternative vote. Викидим (talk) 17:41, 13 April 2024 (UTC)
 * I'd like to see a source for that, because I've never seen any of those systems called "plurality", but if correct, everything in this article would need to be merged into FPTP and this article rewritten from scratch (because it only talks about FPTP right now). –Maximum Limelihood Estimator 17:43, 13 April 2024 (UTC)
 * In general FPTP = SMP (single-member plurality a.k.a. simple plurality). I can point to Maurice Duverger of the Duverger Law fame. A brief summary of his beliefs can be found in not-very-authoritative source here: on p. 331 ("Plurality system with additional conditions"). Sorry, I do not have Duverger's book handy and cannot check his terminology immediately. This is not the only PoV; many authors use FPTP and plurality system synonymously and drop the single-member/simple qualifiers - but many retain them. Викидим (talk) 18:25, 13 April 2024 (UTC)
 * "A brief summary of his beliefs can be found in not-very-authoritative source here: [1 ] on p. 331 ('Plurality system with additional conditions'). Sorry, I do not have Duverger's book handy and cannot check his terminology immediately."
 * The text doesn't seem to quote Duverger himself directly, it just talks about Duverger's law.
 * The text is, as mentioned, a not-very-authoritative source.
 * The text also seems to be confusing the use of a single non-transferable vote in the first round of a runoff (which can arguably be called "plurality voting") with the two-round runoff as a whole, which is not a kind of plurality voting.
 * "This is not the only PoV; many authors use FPTP and plurality system synonymously and drop the single-member/simple qualifiers—but many retain them."
 * The synonymous usage seems to be by far the more common use in social choice theory. With regards to the other two terms you mentioned:
 * "Single-member" means the combination of plurality with single-member districts, i.e. local elections. The term "single-member" is used to distinguish plurality voting from its multi-member (multi-winner) variant, called SNTV.
 * "Simple plurality" has the same meaning as in "simple majority" (which means a majority, ignoring abstentions). In other words, it's a plurality of votes, ignoring abstentions: if the largest number of votes is for "abstain", this option is ignored, rather than leading to a do-over election. –Maximum Limelihood Estimator 15:37, 15 April 2024 (UTC)
 * IMHO for authors that consider plurality synonymous with FPTP, the qualifiers "single-member", "simple" would be excessive and unnecessary, their very use underscores that differences do exist. SNTV seems to me like one more plurality system. Викидим (talk) 19:09, 15 April 2024 (UTC)
 * SNTV is one of several minor variants of FPTP/Plurality, yes, which is why it has a separate article. But as the article itself says in the opening, SNTV is a kind of FPTP.
 * IMHO for authors that consider plurality synonymous with FPTP, the qualifiers "single-member", "simple" would be excessive and unnecessary,
 * Yes, they would be, because this is how everyone understands the term "Plurality"! If the term was this ambiguous, nobody would use it; they'd just say FPTP. As mentioned, the current article says nothing about these other systems, because nobody uses the term plurality like this.
 * In all my years reading the social choice literature I have never seen anyone use "plurality" to mean "instant-runoff voting", and the source you provided is at best ambiguous as to whether it's even doing that. –Maximum Limelihood Estimator 19:54, 15 April 2024 (UTC)
 * I can easily agree with you on instant runoff. For the purposes of this discussion, it is enough to establish that FPTP and "plurality" are not the same thing (SNTV does that), and I am way out of my depth to provide any suggestions on where the evolution of Plurality voting should go. Викидим (talk) 19:58, 15 April 2024 (UTC)
 * For the purposes of this discussion, it is enough to establish that FPTP and "plurality" are not the same thing (SNTV does that),
 * OK, so how about :
 * 1. Redirecting "FPTP" to "Plurality" (which is the most common use), and then
 * 2. Putting up a "For the multi-winner variant of plurality, see SNTV".
 * I am way out of my depth to provide any suggestions on where the evolution of Plurality voting should go.
 * That's fair. –Maximum Limelihood Estimator 20:05, 15 April 2024 (UTC)
 * I am no expert on the issue. However, (1) FPTP is a colloquial, but very well-known - and easy to understand - term. Few people would understand the plurality correctly. Correspondingly, the sheer volume of information on FPTP is much larger than that on other forms of plurality combined. So, if redirecting, it would be expedient and historically correct, to go other way, explaining the non-FPTP plurality as a deviation from (or an improvement of) the simple one. (2) I do not think this would be the best approach. Instead, this article can become an overview of the voting systems that are rightly or wrongly described as plurality systems, with a list of actual use. (3) The main dichotomy, IMO, is between the proportional / plurality systems, and this might be a good place to expound on their differences. Викидим (talk) 20:46, 15 April 2024 (UTC)
 * So, I checked on Google Scholar to be extremely sure. Of the first 20 hits for "plurality voting", 19 were about first-past-the-post in a single-winner context, and one was a false positive (it picked up "plural voting" instead of "plurality voting").
 * To make sure this wasn't just a technical term, I googled "plurality voting". I took the first 5 reliable sources I came across. (I judged one source unreliable, but it agreed as well). Of those 5 pages, 4 defined plurality voting as a single-winner election method (i.e. defined it as FPTP), those being Britannica, Ballotpedia, and two pages from Moravian and Georgetown universities. The 5th was this page. –Sincerely, A Lime 21:58, 13 May 2024 (UTC)
 * Agree, Oppose Arotparaarms (talk) 22:42, 13 April 2024 (UTC)
 * Opppose
 * Plurality voting is a broader concept, which includes:
 * - SNTV - a multi winner system with very different results than FPTP
 * - bloc voting, general ticket, the multi-winner version with similar effects. Because of it's widespread use, FPTP deserves a seperate article which doesn't include these
 * - limited voting, approval voting (of course, not PAV, and of course, even single winner is not often called plurality voting since it would be confusing)
 * it is important to contrast plurality voting to:
 * -majority voting - usually IRV and TRS would be considered so as they have the principle of having next rounds if there is no majority. otherwise, majority and plurality are similar rules, so highlight that not everywhere they have the same linguistic usage - "relative/absolute/simple majority/plurality
 * -proportional representation, completely different principle Rankedchoicevoter (talk) 07:08, 19 April 2024 (UTC)
 * That said, if you're right, we'll need to merge this page into FPTP instead of the other way around, and then turn this page into a disambiguation directing people to choose SNTV or FPTP. –Maximum Limelihood Estimator 16:39, 24 April 2024 (UTC)

@Rankedchoicevoter the term "Majority voting" in social choice theory has a much more restrictive meaning (see Condorcet winner criterion, often called majority-rule in the social choice/voting theory/welfare economics literature). IRV and two-round would be considered to fall under the sequential-loser or core support umbrella instead, alongside plurality.

Approval voting would never be considered a variant on plurality voting. (On the other hand, TRS might actually be counted as a variant of FPP/Plurality.)

"Plurality" without qualification excludes all the systems you mentioned. FPTP also includes SNTV (the proportional-representation form of FPTP). The phrases FPTP and Plurality voting are synonymous. –Maximum Limelihood Estimator 23:29, 23 April 2024 (UTC)

Terminology sources
Trying to list the works that can be used as references for terminology. Feel free to expand or object. Викидим (talk) 01:54, 24 April 2024 (UTC)
 * Recommended in

 * It looks like Rosen (1974) says:
 * "Bloc voting and plurality voting are, respectively, the multi-member and single-member forms of the first-past-the-post class of systems."
 * Nice source, I like it :) –Sincerely, A Lime 21:45, 13 May 2024 (UTC)
 * Good. I have no deep knowledge, so any agreement on the sources would work for me. It is clear to me by now that the terminology varies across the researchers; this is not an unusual situation for many fields of study. We can simply accept someone's terminology (with attribution), and mention other versions. Ideally, we should find a source that acknowledges the differences and attributes them to particular researchers. While writing Zigzag Moderne (about an architectural style the very existence of which is debated), I was lucky to have such a source. Let's hope we can find one here, too. Викидим (talk) 23:32, 13 May 2024 (UTC)
 * That sounds like the other way way around. Plurality voting is the general type, bloc voting a multi-member version. I added sources in the article supporting that view. I am aware of the wide range of use for these terms and when it gets more rigorous, these have to be clarified, but first-past-the-post is same as SMP, and it would add additional confusion to say otherwise (it's in the name "first", at least that part makes sense, even if there is no post...). Rankedchoicevoter (talk) 09:07, 14 May 2024 (UTC)
 * As I have said, I am OK with any arrangement that would state something like "X defines PV as a subset of FPTP[cite]" (or the reverse). It seems to me that there is no consistency here, so we need to attribute the definitions. Викидим (talk) 09:39, 14 May 2024 (UTC)
 * "First" can also just mean the beginning of a list, e.g. "the first 100 people to sign up will receive a discount".
 * FWIW, I agree that I don't like the use of the term "FPTP" as including SNTV; if I wanted to say SNTV, I'd say SNTV. It seems like the vast majority of papers nowadays use "Plurality" and "FPTP" interchangeably, but some older sources try to maintain a distinction; but this is inconsistent. –Sincerely, A Lime 20:44, 14 May 2024 (UTC)
 * If this is the case, we need to reflect it. It would be great to find a source that reflects this change. Викидим (talk) 00:15, 15 May 2024 (UTC)
 * A good example of people using "Plurality" quite differently from how most people use it today is Merrill and Nagel 1987, who use it to mean any voting rule that elects the candidate(s) with the highest score(s). (Which makes ~no sense to me, because every voting rule does that.) –Sincerely, A Lime 02:18, 26 May 2024 (UTC)
 * (From what I can tell everyone else agrees, because I was unable to find any other papers using this meaning.) –Sincerely, A Lime 02:19, 26 May 2024 (UTC)
 * It makes perfect sense to me. The proportional voting does not elect candidates with most of the votes, it elects a particular number of representatives from party lists, while parties decide themselves (ahead of the elections) who the representatives will be. Merrill and Nagel are not alone in the main (top-level) classification split being proportional vs. plurality. These researchers treat plurality a a generic case, with further details like FPTP vs. multi-candidate districts. Викидим (talk) 03:36, 26 May 2024 (UTC)
 * Well, the thing is, you can define proportional representation as being plurality rules in that sense too... see highest averages method: Seats are awarded to the candidate with the highest value of $$\frac{\text{votes}}{\text{seats}}$$ (or a slightly different ratio for different divisor methods).
 * No matter what you do, it's still a "plurality" rule :)
 * I think it's supposed to contrast with a majority in the sense that, if no candidate wins a majority, there is no winner. Balloting continues until a candidate achieves a majority. Most parliamentary officers are elected this way (e.g. the Speaker of the House). –Sincerely, A Lime 04:00, 26 May 2024 (UTC)
 * All I am trying to say is that a significant number of researchers appear to split the voting methods into two broad groups: proportional vs. plurality, with further subdivision happening on the inside of these two classes. Whether they are right or wrong is immaterial, IMHO: we simply need to reflect this PoV (part of the mainstream) somewhere. To me, this article is a convenient place to do so: if some expert takes an opinion that "plurality" is a synonym of FPTP, we can mention their position here and write down the details in the FPTP article. If, however, an expert thinks that plurality is the opposite of proportionality, their opinion belongs here. Викидим (talk) 04:56, 26 May 2024 (UTC)
 * Right, what I'm saying is I don't think any expert defines "plurality" as just meaning "not proportional".
 * In any case I'd give this glossary as a decent citation (where it defines "plurality" to mean FPTP).
 * http://ndl.ethernet.edu.et/bitstream/123456789/77460/1/21.pdf#page=138 –Sincerely, A Lime 16:51, 28 May 2024 (UTC)
 * (1) I am happy with this text. (2) However, it also says that the multiple-candidates election is also plurality, so plurality is not equal to FPTP. (3) There are plenty of sources that are published in peer-review journals and books from reputed publishers that contain very clear language of opposition between PR and PV, like "plurality vs proportional", etc. (4) Therefore, there is no single definition that we can present to the reader, IMHO. Ideally, it would be great to find a review of contradictions in definitions. Викидим (talk) 18:12, 28 May 2024 (UTC)
 * Here is one of these sources with very clear language: on page 3. Викидим (talk) 18:14, 28 May 2024 (UTC)
 * Interesting, thanks. I'd never seen that before. I think this explains the disagreement and confusion:
 * 1. Some (particularly older) sources, mostly in political science, define "plurality voting" to mean winner-take-all representation.
 * 2. Other (especially newer) sources, including most of social choice, define "plurality" as an abbreviation of the term first-preference plurality.
 * In that case, is it reasonable to make "plurality voting" a disambiguation page that explains these two common meanings? cc @Rankedchoicevoter Closed Limelike Curves (talk) 18:43, 14 June 2024 (UTC)
 * My goal here is twofold:
 * I am sure there is a conflict of definitions even in the expert literature. We need to reflect this conflict in our encyclopedia, so the readers can understand better both other articles in Wikipedia and the external literature
 * In order to achieve #1, we should not merge plurality and FPTP, IMHO
 * Викидим (talk) 00:48, 15 June 2024 (UTC)

title: Talk:Plurality voting


text:
Robert Estienne

Robert I Estienne (1503 – 7 September 1559), known as Robertus Stephanus in Latin and sometimes referred to as Robert Stephens, was a 16th-century printer in Paris. He was the proprietor of the Estienne print shop after the death of his father Henri Estienne, the founder of the Estienne printing firm. Estienne published and republished many classical texts as well as Greek and Latin translations of the Bible. Known as "Printer to the King" in Latin, Hebrew, and Greek, Estienne's most prominent work was the Thesaurus linguae latinae which is considered to be the foundation of modern Latin lexicography. Additionally, he was the first to print the New Testament divided into standard numbered verses.

Raised a Catholic, he became a Protestant late in his life. Many of his published Bibles included commentary which upset the Catholic theologians of the Sorbonne who sought to censor Estienne's work. Eventually, overcome by the prejudice of the Sorbonne, Estienne and his family fled to Geneva where he continued his printing uncensored, publishing many of the works of John Calvin. In 1556 he became a citizen of Geneva, where he would die on 7 September 1559.

Of Estienne's four sons, two became accomplished printers, one of whom was Henri Estienne who continued the legacy of his grandfather Estienne's printing firm. Along with other printers, Estienne contributed to the "Golden Age of French Typography".

Early life
Robert Estienne was born in Paris in 1503. The second son of the famous humanist printer Henri Estienne, he became knowledgeable in Latin, Greek, and Hebrew. After his father's death in 1520, the Estienne printing establishment was maintained by his father's former partner Simon de Colines who also married Estienne's mother, the widow Estienne. As Estienne was not yet of age at the time of his father's death, Colines and Gilles Nepveu (the husband or fiancé of his sister Nicole) became his legal guardians. Estienne and Colines likely collaborated in Estienne print shop for a time. Colines was known for his exquisite type cutting, whereas Estienne was known for his accuracy.

In 1526, Robert Estienne assumed control of his father's printing shop while Colines established his own firm nearby. The parties agreed to divide the house, printing equipment, and printing supplies in half. Colines moved his shop down the street from the Estienne shop. Though the nature of their relationship after this is largely unknown, scholars suggest that they had mutual respect for one another and may have continued to collaborate, sharing fonts and materials. Even though Estienne re-established his father's printing shop in 1526, his first independent project as a scholar-printer can be traced back to 1524. He was in the process of publishing a Latin version of the Bible as he searched Paris for manuscripts. He had already printed a New Testament, and some slight alterations which he had introduced into the text brought upon him the censures of the faculty of theology. It was the first of a long series of disputes between him and that body. Around this time, he apparently joined the Reformed Church.

Estienne married Perrette Badius in 1526 whose father Josse Badius Ascensius owned a print shop, giving Estienne the resources to print that he lacked from his father's materials alone. After her father's death in 1535, Estienne merged the Estienne and Badius printing business. His first Biblia or version of the Vulgate Bible was published in 1527. While he was working on the Bible, he increased his revenue and reputation by publishing a series of octavos, which in this case were small, inexpensive educational books from scholars such as Cicero and Lucian. Estienne's trade was primarily as printer-bookseller and though he did publish his own prints, he did not publish in partnership with other printers as was a common practice for printers of the time.

Latin classics
With his title of "royal typographer" Robert Estienne promoted the Estienne print shop by his numerous editions of grammatical works and other schoolbooks (among them many of Melanchthon's) and of classical and Patristic authors, such as Dio Cassius, Cicero, Sallust, Julius Caesar, Justin, Socrates Scholasticus, and Sozomen. During the first fifteen years of his career, Estienne focused his printing on five Latin classic authors, specifically, Cicero, Terence, Plautus, Pliny, and Virgil. He printed works from Horace and Persisus, but he printed them far less frequently. He nearly tripled the number of authors' works he published from 1541 to 1545. Scholars suggest that Estienne's trouble with his published Bibles and the Catholic Church led him to publish more authors of Latin Classics as a buffer.

Many of Estienne's published classics, especially the Greek editions (which were printed with typefaces made by Claude Garamond), were famous for their typographical elegance. The editiones principes issued from Estienne's press were eight in number. He began with the Historia ecclesiastica (1544) and ended with Appian (1551). The last was completed after Estienne's departure from Paris by his brother Charles and appeared under Charles's name. Estienne also printed numerous editions of Latin classics, of which the folio Virgil of 1532 is the most noteworthy. He printed a large number of Latin grammars and other educational works, many of which were written by Mathurin Cordier, his friend and co-worker in the cause of humanism. He was trained as a punchcutter, but no font has been identified as his. Estienne did, however, oversee the work of the best punchcutters of the time such as Claude Garamond and Guillaume Le Bé. Under Estienne, Garamond designed the Greek type used by the King of France which was used to print the first edition of Roman History. Consequently, Estienne was the first printer granted permission to use the grecs du roi or Greek types of the king. In the 1530s Estienne's printing represents the first use of apostrophes and grave and acute accents in France. Moreover, Estienne was known as one of the printers responsible for adapting the Aldine roman type in France.

Dictionaries
One of the best printers of his time, Robert Estienne was asked to either compile a dictionary from the best Latin authors or make one himself; in 1531 he published Thesaurus linguae latinae, which is considered by some scholars to be the foundation of modern Latin lexicography. Moreover, this dictionary made Estienne the "father of French lexicography". He had worked on it for two years, with the assistance only of Thierry of Beauvais. It was 964 pages and was improved in 1536 and 1543 in three volumes. Considered his "greatest monument of Latin scholarship", he employed research assistants for the 1543 version: Andreas Gruntleus, Gerardus Clericus, and Adam Nodius. From his work on the Thesaurus linguae latinae, he published Dictionarium latino-gallicum in 1538 and Dictionaire francoislatin in 1540. These dictionaries were superior to others at the time because non-classical elements had been edited out; when determining words, they were checked for correctness and applicability in context; and citations were exclusively from classical authors. Furthermore, it applied consistency to word order since lexicographers disagreed about whether words should be ordered alphabetically or etymologically. Estienne's thesaurus was alphabetical based on the first three letters of the word, then grouped etymologically. In the 1540s, he began publishing more concise school dictionaries. Many of these dictionaries were translated into other languages such as German and Flemish. From 1528 to 1580, he published several editions of Alphabetum graecum, a representation of Renaissance Greek orthography.

Religious texts
Although Robert Estienne was printing Bibles in Latin as early as 1528, he printed his first Greek New Testament in 1546. Despite its similarity to the works of Erasmus, Estienne did not credit Erasmus and rather claimed to be influenced by ancient codices. The first two are beautiful Greek texts, called O mirifica. The third and most significant is known as the Editio Regia or the "Royal Edition", published in 1550 for King Henri II. Typographically it is significant because the Greek font made by Garamond became the most widely used Greek font for European printers. In it he combined over 15 Greek sources with annotations in the margins. The 1550 version became known as the Textus Receptus, the standard text for many generations. The 1551 edition contains Erasmus's Latin translation and the Vulgate. Scholars have described his editing of the Vulgate as mediocre and lacking in effort or depth. In this edition of the Vulgate, Estienne introduced the division of the New Testament into chapter and verses for the first time. After he finished the Vulgate, he began developing his style. He was interested in working on original texts rather than translations. Additionally, he was interested in writing commentaries to help an average reader understand the academic texts to the point of adding his own interpretation. Moreover, Estienne's commentary in the fourth edition of the Greek New Testament initiated the antagonism of the Sorbonne against Estienne.

He published two editions of the Hebrew Bible: one in 13 volumes and another one in 10 volumes. Estienne acquired Vulgate manuscripts while in Paris and printed a number of editions throughout his career. The principal editions are the 1527, 1532, 1540 (one of the ornaments of his press), and 1546. In the 1532 edition, he placed the Acts in between the gospels and epistles of Paul as is standard in most Bibles. Before this, the Acts were usually found at the end. Furthermore, typographer and printing historian Stanley Morison claimed that Estienne's 1532 folio Bible contained, "what is probably the finest use ever made of [the Garamond] letter." Estienne printed this edition of the Bible in a grand folio format; his expected buyers were the nobility and the wealthy rather than the university faculty/students. Though in 1543, his style shifted to that of sextodecimo format, printing Bibles in Latin, Greek, and Hebrew, which assumes the buyers are students and professors.

His editions, especially that of 1546, containing a new translation at the side of the Vulgate, was the subject of sharp and acrimonious criticism from the clergy. In 1539 he received the distinguishing title of "Printer to the king" for Latin and Hebrew, and later for Greek. This incited anger from the Sorbonne because Estienne had converted from Catholicism to Protestantism. The Sorbonne was opposed to the humanist ideals of the time and was attempting to censor Estienne's publishing firm. He was protected by Francis I of France with whom he enjoyed strong patronage and friendship; Estienne aided Francis I in printing documents ratifying policies which established and justified his power. Later, Estienne published a document to inform the public how alliances between French royalty, German Protestants, and Turkish royalty were beneficial for European religious peace. In 1538, Francis I requested that Estienne give a copy of every Greek book he had printed to create the royal library, which became the first copyright library. However, after Francis I died in 1547 and was succeeded by Henry II, Estienne fled to Geneva around 1550. With him, he brought his printing material, including his Greek type made by Garamond.

Geneva
On his arrival at Geneva, Robert Estienne published a defense against the attacks of the Sorbonne in 1552 called his Réponse. It was first published in Latin. He later translated it into French and published it again. The central theme of his Réponse is that the Sorbonne had a great ability to persuade or intimidate people. Estienne established his printing firm in Geneva and his brother Charles helped run the firm in Paris. However, after Charles died in a debtors' prison, Robert II (the son of Robert I) took over the business. In Geneva, Estienne issued the French Bible in 1553 and many of John Calvin's writings, including the Institutio in 1553. His 1556 edition of the Latin Bible contained the translation of the Old Testament by Santes Pagninus and the first edition of Theodore Beza's Latin edition of the New Testament. In 1556 he became a citizen of the Republic of Geneva, where he died on 7 September 1559. Estienne's other sons, Henri II and François, helped Estienne run the shop in Geneva after the death of Estienne. Robert Estienne was one of the most successful printers in the Estienne family, and one of the best scholars of the time. It was in part, due to Estienne that the reign of Francis I was considered the "Golden Age of French Typography."

Pressmarks
Robert Estienne used several pressmarks or devices on his prints. Estienne's pressmark with an olive branch and a serpent wound around a spear was first seen in 1544 on the title page of Preparatio Evangelica. It was symbolic of wisdom in times of war and peace. The motto below it translates to "to the wise king and the valiant warrior". Another device was called Oliva Stephanorum or the olive of the Stephens family with the words of Romans 11:20, Noli altum sapere ("Do not be proud") and later Noli altum sapere, sed time ... ("Be not high-minded but fear"). The device shows a man standing by an olive tree. Scholars believe this man is Paul the Apostle who is affirming the importance of faith. This is consistent with Estienne's connection to the Protestant Reformation. The olive tree is meant to represent the tree of knowledge. The device may have been a subtle attack on the Catholic theologians at the Sorbonne for their "lack of humility". Pressmarks function best when they are immediately recognized, and scholars criticize Estienne's pressmarks for not being easily recognizable.

Family
Robert Estienne encouraged his four sons to study and perfect his professions. His will indicated that he wished all of his sons follow in his profession. Two of Robert's sons, Henri and Robert became successful printers. François (born 1540) printed in Geneva from 1562 to 1582. As well as issuing editions of the Bible in Latin and French, he published some of Calvin's works.

Robert Estienne II (1530–1570) studied Hebrew as his father recommended. Uninterested in the Reformation, he stayed in Paris instead of following his father to Geneva, opening his own printing shop in 1556. He earned the title of Typographus regius in 1563. He printed the New Testament of 1568–1569, a reprint of his father's first edition. He printed the Decalogue in Hebrew and Aramaic in 1566. Additionally, Estienne printed books in Hebrew for professors in Paris, but fled to Geneva in 1569, because he worked for Anglican clients. He died in 1570.

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title: Robert Estienne


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Henri Estienne

Henri Estienne (, ; 1528 or 1531 – 1598), also known as Henricus Stephanus , was a French printer and classical scholar. He was the eldest son of Robert Estienne. He was instructed in Latin, Greek, and Hebrew by his father and would eventually take over the Estienne printing firm which his father owned in 1559 when his father died. His most well-known work was the Thesaurus graecae linguae, which was printed in five volumes. The basis of Greek lexicology, no thesaurus would rival that of Estienne's for three hundred years.

Among his many publications of Greek authors, his publications of Plato are the source of Stephanus pagination, which is still used to refer to Plato's works. Estienne died in Lyon in 1598.

Life
Henri Estienne was born in Paris in 1528 or 1531. His father instructed him in Latin, Greek, Hebrew, and typography, and according to a note in his edition of Aulus Gellius (1585), he picked up some Latin as a child, as that language was used as a lingua franca in the multi-national household. However, he was primarily instructed in Greek by Pierre Danès. He was also educated by other French scholars such as Adrianus Turnebus. He began working for his father's business at age eighteen and was employed by his father to collate a manuscript of Dionysius of Halicarnassus. In 1547, as part of his training, he traveled to Italy, England, and Flanders, where he learned Spanish and busied himself in collecting and collating manuscripts for his father's press. Around 1551, Robert Estienne fled to Geneva with his family, including Henri Estienne, to escape religious persecution in Paris. The same year, he translated Calvin's catechism into Greek, which was printed in 1554 in his father's printing room.

Estienne published the Anacreon in 1554, which was his first independent work. Afterwards, he returned to Italy to assist the Aldine Press in Venice. In Italy, he discovered a copy of Diodorus Siculus in Rome, and returned to Geneva in 1555. In 1557 he likely had a printing establishment of his own, advertising himself as the "Parisian printer" (typographus parisiensis). The following year he assumed the title illustris viri Huldrici Fuggeri typographus from his patron, Ulrich Fugger who saved him from financial despair after the death of his father. Estienne published the first anthology that included sections from Parmenides, Empedocles, and other Pre-Socratic philosophers. In 1559, on his father's death, Estienne assumed charge of his presses and became Printer of the Republic of Geneva. In the same year he produced his own Latin translation of the works of Sextus Empiricus, and an edition of Diodorus Siculus based on his earlier discoveries. In 1565, he printed a large French Bible. The following year he published his best-known French work, the Apologie pour Hérodote. Some passages being considered objectionable by the Geneva consistory, he was compelled to cancel the pages containing them. The book became highly popular, and within sixteen years twelve editions were printed. Estienne used the type he inherited and did not invent any new types.

His most celebrated work, the Thesaurus graecae linguae or Greek thesaurus, appeared in five volumes in 1572. This thesaurus was a sequel to Robert Estienne's Latin thesaurus. The basis of Greek lexicography, a Greek thesaurus to rival that of Estiennes was not printed for over 300 years. This work was begun by his father and served up to the nineteenth century as the basis of Greek lexicography. However, the sale of this thesaurus was impeded due to its high price and the printing of an abridged copy later. In 1576 and 1587, Estienne published two Greek versions of the New Testament. The 1576 version contained the first scientific treatise on the language of the apostolic writers. The 1587 version contained a discussion on the ancient divisions of the text. Estienne's other publications included those of Herodotus, Plato, Horace, Virgil, Plutarch, and Pliny the Elder. He also published an edition of Aeschylus, in which Agamemnon was printed in its entirety and as a separate play for the first time.

In 1578 he published the first and one of the most important editions of the complete works of Plato, translated by Jean de Serres, with commentary. This work is the source of the standard "Stephanus numbers" used by scholars today to refer to the works of Plato.

The publication in 1578 of his Deux Dialogues du nouveau françois italianizé brought him into a fresh dispute with the consistory. To avoid their censure he went to Paris, and resided at the French court for a year. On his return to Geneva he was summoned before the consistory and was imprisoned for a week. From this time his life became more and more nomadic. He traveled to Basel, Heidelberg, Vienna, and Pest. He also spent time in Paris and other regions in France. These journeys were undertaken partly in the hope of procuring patrons and purchasers, for the large sums which he had spent on such publications as the Thesaurus and the Plato of 1578 had almost ruined him. He published a concordance of the New Testament in 1594.

After visiting the University of Montpellier, where Isaac Casaubon, his son-in-law, was now professor, he started for Paris. He was taken ill in Lyon, and died there at the end of January 1598.

Family
Henri Estienne was married three times. He married Marguerite Pillot in 1555, Barbe de Wille in 1556, and Abigail Pouppart in 1586. Estienne had fourteen children, three of whom survived him. His daughter was married to Isaac Casaubon. His son Paul (born 1567) assumed control of the presses in Geneva with Casaubon but he fled to Paris from the authorities. Paul's son Antoine became "Printer to the King" in Paris and "Guardian of the Greek Matrices"; however his death in 1674 ended the nearly two-century-long Estienne printing business.

Legacy
Henri Estienne is considered by some scholars to be the most prominent printer in the Estienne family. Estienne was one of the "greatest and last scholarly editors and publishers of the Renaissance."

title: Henri Estienne


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Central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.

The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory.

An elementary form of the theorem states the following. Let $$X_1, X_2, \dots, X_n$$ denote a random sample of $$n$$ independent observations from a population with overall expected value (average) $$\mu$$ and finite variance $$\sigma^2$$, and let $$\bar{X}_n$$ denote the sample mean of that sample (which is itself a random variable). Then the limit as $n\to\infty$ of the distribution of $$(\bar{X}_n-\mu)/\sigma_{\bar{X}_n},$$ where $$\sigma_{\bar{X}_n}=\sigma / \sqrt{n},$$ is the standard normal distribution.

In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and that the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.

The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.

Classical CLT
Let $$\{X_1, \ldots, X_n}\$$ be a sequence of i.i.d. random variables having a distribution with expected value given by $$\mu$$ and finite variance given by $$\sigma^2.$$ Suppose we are interested in the sample average $$\bar{X}_n \equiv \frac{X_1 + \cdots + X_n}{n}.$$

By the law of large numbers, the sample average converges almost surely (and therefore also converges in probability) to the expected value $$\mu$$ as $$n\to\infty.$$

The classical central limit theorem describes the size and the distributional form of the stochastic fluctuations around the deterministic number $$\mu$$ during this convergence. More precisely, it states that as $$n$$ gets larger, the distribution of the difference between the sample average $$\bar{X}_n$$ and its limit $$\mu,$$ when multiplied by the factor $$\sqrt{n}$$ — that is, $$\sqrt{n}(\bar{X}_n - \mu)$$ — approaches the normal distribution with mean $$0$$ and variance $$\sigma^2.$$ For large enough $$n,$$ the distribution of $$\bar{X}_n$$ gets arbitrarily close to the normal distribution with mean $$\mu$$ and variance $$\sigma^2/n.$$

The usefulness of the theorem is that the distribution of $$\sqrt{n}(\bar{X}_n - \mu)$$ approaches normality regardless of the shape of the distribution of the individual $$X_i.$$ Formally, the theorem can be stated as follows:

$$

In the case $$\sigma > 0,$$ convergence in distribution means that the cumulative distribution functions of $$\sqrt{n}(\bar{X}_n - \mu)$$ converge pointwise to the cdf of the $$\mathcal{N}(0, \sigma^2)$$ distribution: for every real number $$z,$$ $$\lim_{n\to\infty} \mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] = \lim_{n\to\infty} \mathbb{P}\left[\frac{\sqrt{n}(\bar{X}_n-\mu)}{\sigma } \le \frac{z}{\sigma}\right]= \Phi\left(\frac{z}{\sigma}\right) ,$$ where $$\Phi(z)$$ is the standard normal cdf evaluated at $$z.$$ The convergence is uniform in $$z$$ in the sense that $$\lim_{n\to\infty}\;\sup_{z\in\R}\;\left|\mathbb{P}\left[\sqrt{n}(\bar{X}_n-\mu) \le z\right] - \Phi\left(\frac{z}{\sigma}\right)\right| = 0~,$$ where $$\sup$$ denotes the least upper bound (or supremum) of the set.

Lyapunov CLT
In this variant of the central limit theorem the random variables $X_i$ have to be independent, but not necessarily identically distributed. The theorem also requires that random variables $\left| X_i\right|$ have moments of some order $(2+\delta)$, and that the rate of growth of these moments is limited by the Lyapunov condition given below.

$$

In practice it is usually easiest to check Lyapunov's condition for $\delta = 1$.

If a sequence of random variables satisfies Lyapunov's condition, then it also satisfies Lindeberg's condition. The converse implication, however, does not hold.

Lindeberg CLT
In the same setting and with the same notation as above, the Lyapunov condition can be replaced with the following weaker one (from Lindeberg in 1920).

Suppose that for every $\varepsilon > 0$ $$ \lim_{n \to \infty} \frac{1}{s_n^2}\sum_{i = 1}^{n} \operatorname E\left[(X_i - \mu_i)^2 \cdot \mathbf{1}_{\left\{\left| X_i - \mu_i \right| > \varepsilon s_n \right\}} \right] = 0$$ where $\mathbf{1}_{\{\ldots\}}$ is the indicator function. Then the distribution of the standardized sums $$\frac{1}{s_n}\sum_{i = 1}^n \left( X_i - \mu_i \right)$$ converges towards the standard normal distribution $\mathcal{N}(0, 1)$.

Multidimensional CLT
Proofs that use characteristic functions can be extended to cases where each individual $\mathbf{X}_i$ is a random vector in $\R^k$, with mean vector $\boldsymbol\mu = \operatorname E[\mathbf{X}_i]$  and covariance matrix $\mathbf{\Sigma}$  (among the components of the vector), and these random vectors are independent and identically distributed. The multidimensional central limit theorem states that when scaled, sums converge to a multivariate normal distribution. Summation of these vectors is done component-wise.

For $$i = 1, 2, 3, \ldots,$$ let $$\mathbf{X}_i = \begin{bmatrix} X_{i}^{(1)} \\ \vdots \\ X_{i}^{(k)} \end{bmatrix}$$ be independent random vectors. The sum of the random vectors $$\mathbf{X}_1, \ldots, \mathbf{X}_n$$ is $$\sum_{i=1}^{n} \mathbf{X}_i = \begin{bmatrix} X_{1}^{(1)} \\ \vdots \\ X_{1}^{(k)} \end{bmatrix} + \begin{bmatrix} X_{2}^{(1)} \\ \vdots \\ X_{2}^{(k)} \end{bmatrix} + \cdots + \begin{bmatrix} X_{n}^{(1)} \\ \vdots \\ X_{n}^{(k)} \end{bmatrix} = \begin{bmatrix} \sum_{i=1}^{n} X_{i}^{(1)} \\ \vdots \\ \sum_{i=1}^{n} X_{i}^{(k)} \end{bmatrix}$$ and their average is $$\mathbf{\bar X_n} = \begin{bmatrix} \bar X_{i}^{(1)} \\ \vdots \\ \bar X_{i}^{(k)} \end{bmatrix} = \frac{1}{n} \sum_{i=1}^{n} \mathbf{X}_i.$$ Therefore, $$\frac{1}{\sqrt{n}} \sum_{i=1}^{n} \left[ \mathbf{X}_i - \operatorname E \left( \mathbf{X}_i \right) \right] = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} ( \mathbf{X}_i - \boldsymbol\mu ) = \sqrt{n}\left(\overline{\mathbf{X}}_n - \boldsymbol\mu\right). $$

The multivariate central limit theorem states that $$\sqrt{n}\left( \overline{\mathbf{X}}_n - \boldsymbol\mu \right) \mathrel{\overset{d}{\longrightarrow}} \mathcal{N}_k(0,\boldsymbol\Sigma),$$ where the covariance matrix $$\boldsymbol{\Sigma}$$ is equal to $$ \boldsymbol\Sigma = \begin{bmatrix} {\operatorname{Var} \left (X_{1}^{(1)} \right)} & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(2)} \right) & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1}^{(1)},X_{1}^{(k)} \right) \\ \operatorname{Cov} \left (X_{1}^{(2)},X_{1}^{(1)} \right) & \operatorname{Var} \left( X_{1}^{(2)} \right) & \operatorname{Cov} \left(X_{1}^{(2)},X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left(X_{1}^{(2)},X_{1}^{(k)} \right) \\ \operatorname{Cov}\left (X_{1}^{(3)},X_{1}^{(1)} \right) & \operatorname{Cov} \left (X_{1}^{(3)},X_{1}^{(2)} \right) & \operatorname{Var} \left (X_{1}^{(3)} \right) & \cdots & \operatorname{Cov} \left (X_{1}^{(3)},X_{1}^{(k)} \right) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(1)} \right) & \operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(2)} \right) & \operatorname{Cov} \left (X_{1}^{(k)},X_{1}^{(3)} \right) & \cdots & \operatorname{Var} \left (X_{1}^{(k)} \right) \\ \end{bmatrix}~.$$

The multivariate central limit theorem can be proved using the Cramér–Wold theorem.

The rate of convergence is given by the following Berry–Esseen type result:

$$

It is unknown whether the factor $d^{1/4}$ is necessary.

The Generalized Central Limit Theorem
The Generalized Central Limit Theorem (GCLT) was an effort of multiple mathematicians (Bernstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. The first published complete proof of the GCLT was in 1937 by Paul Lévy in French. An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.

The statement of the GCLT is as follows:

 * A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X1, X2, X3, ... and constants an > 0, bn ∈ ℝ with

 * an (X1 + ... + Xn) − bn → Z.

 * Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy Fn(y) → F(y) at all continuity points of F.

In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a stable distribution.

CLT under weak dependence
A useful generalization of a sequence of independent, identically distributed random variables is a mixing random process in discrete time; "mixing" means, roughly, that random variables temporally far apart from one another are nearly independent. Several kinds of mixing are used in ergodic theory and probability theory. See especially strong mixing (also called α-mixing) defined by $\alpha(n) \to 0$ where $\alpha(n)$  is so-called strong mixing coefficient.

A simplified formulation of the central limit theorem under strong mixing is:

$$

In fact, $$\sigma^2 = \operatorname E\left(X_1^2\right) + 2 \sum_{k=1}^{\infty} \operatorname E\left(X_1 X_{1+k}\right),$$ where the series converges absolutely.

The assumption $\sigma \ne 0$ cannot be omitted, since the asymptotic normality fails for $X_n = Y_n - Y_{n-1}$  where $Y_n$  are another stationary sequence.

There is a stronger version of the theorem: the assumption $\operatorname E\left[X_n^{12}\right] < \infty$ is replaced with $\operatorname E\left[{\left and the assumption $\alpha_n = O\left(n^{-5}\right) $  is replaced with $$\sum_n \alpha_n^{\frac\delta{2(2+\delta)}} < \infty.$$

Existence of such $\delta > 0$ ensures the conclusion. For encyclopedic treatment of limit theorems under mixing conditions see.

Martingale difference CLT
$$

Proof of classical CLT
The central limit theorem has a proof using characteristic functions. It is similar to the proof of the (weak) law of large numbers.

Assume $\{X_1, \ldots, X_n, \ldots \}$ are independent and identically distributed random variables, each with mean $\mu$  and finite variance $\sigma^2$. The sum $X_1 + \cdots + X_n$ has mean $n\mu$  and variance $n\sigma^2$. Consider the random variable $$Z_n = \frac{X_1+\cdots+X_n - n \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{X_i - \mu}{\sqrt{n \sigma^2}} = \sum_{i=1}^n \frac{1}{\sqrt{n}} Y_i,$$ where in the last step we defined the new random variables $Y_i = \frac{X_i - \mu}{\sigma} $, each with zero mean and unit variance ($\operatorname{var}(Y) = 1$ ). The characteristic function of $Z_n$ is given by $$\varphi_{Z_n}\!(t) = \varphi_{\sum_{i=1}^n {\frac{1}{\sqrt{n}}Y_i}}\!(t) \ =\ \varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right) \varphi_{Y_2}\!\! \left(\frac{t}{\sqrt{n}}\right)\cdots \varphi_{Y_n}\!\! \left(\frac{t}{\sqrt{n}}\right) \ =\ \left[\varphi_{Y_1}\!\!\left(\frac{t}{\sqrt{n}}\right)\right]^n, $$ where in the last step we used the fact that all of the $Y_i$ are identically distributed. The characteristic function of $Y_1$ is, by Taylor's theorem, $$\varphi_{Y_1}\!\left(\frac{t}{\sqrt{n}}\right) = 1 - \frac{t^2}{2n} + o\!\left(\frac{t^2}{n}\right), \quad \left(\frac{t}{\sqrt{n}}\right) \to 0$$ where $o(t^2 / n)$ is "little $o$ notation" for some function of $t$  that goes to zero more rapidly than $t^2 / n$. By the limit of the exponential function ($e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n$ ), the characteristic function of $$Z_n$$ equals $$\varphi_{Z_n}(t) = \left(1 - \frac{t^2}{2n} + o\left(\frac{t^2}{n}\right) \right)^n \rightarrow e^{-\frac{1}{2} t^2}, \quad n \to \infty.$$

All of the higher order terms vanish in the limit $n\to\infty$. The right hand side equals the characteristic function of a standard normal distribution $\mathcal{N}(0, 1)$, which implies through Lévy's continuity theorem that the distribution of $Z_n$ will approach $\mathcal{N}(0,1)$  as $n\to\infty$. Therefore, the sample average $$\bar{X}_n = \frac{X_1+\cdots+X_n}{n}$$ is such that $$\frac{\sqrt{n}}{\sigma}(\bar{X}_n - \mu) = Z_n$$ converges to the normal distribution $\mathcal{N}(0, 1)$, from which the central limit theorem follows.

Convergence to the limit
The central limit theorem gives only an asymptotic distribution. As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.

The convergence in the central limit theorem is uniform because the limiting cumulative distribution function is continuous. If the third central moment $\operatorname{E}\left[(X_1 - \mu)^3\right]$ exists and is finite, then the speed of convergence is at least on the order of $1 / \sqrt{n}$  (see Berry–Esseen theorem). Stein's method can be used not only to prove the central limit theorem, but also to provide bounds on the rates of convergence for selected metrics.

The convergence to the normal distribution is monotonic, in the sense that the entropy of $Z_n$ increases monotonically to that of the normal distribution.

The central limit theorem applies in particular to sums of independent and identically distributed discrete random variables. A sum of discrete random variables is still a discrete random variable, so that we are confronted with a sequence of discrete random variables whose cumulative probability distribution function converges towards a cumulative probability distribution function corresponding to a continuous variable (namely that of the normal distribution). This means that if we build a histogram of the realizations of the sum of $n$ independent identical discrete variables, the piecewise-linear curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as $n$ approaches infinity; this relation is known as de Moivre–Laplace theorem. The binomial distribution article details such an application of the central limit theorem in the simple case of a discrete variable taking only two possible values.

Common misconceptions
Studies have shown that the central limit theorem is subject to several common but serious misconceptions, some of which appear in widely used textbooks. These include:
 * The misconceived belief that the theorem applies to random sampling of any variable, rather than to the mean values (or sums) of iid random variables extracted from a population by repeated sampling. That is, the theorem assumes the random sampling produces a sampling distribution formed from different values of means (or sums) of such random variables.
 * The misconceived belief that the theorem ensures that random sampling leads to the emergence of a normal distribution for sufficiently large samples of any random variable, regardless of the population distribution. In reality, such sampling asymptotically reproduces the properties of the population, an intuitive result underpinned by the Glivenko-Cantelli theorem.
 * The misconceived belief that the theorem leads to a good approximation of a normal distribution for sample sizes greater than around 30, allowing reliable inferences regardless of the nature of the population. In reality, this empirical rule of thumb has no valid justification, and can lead to seriously flawed inferences. See Z-test for where the approximation holds.

Relation to the law of large numbers
The law of large numbers as well as the central limit theorem are partial solutions to a general problem: "What is the limiting behavior of $n → ∞$ as $n$ approaches infinity?" In mathematical analysis, asymptotic series are one of the most popular tools employed to approach such questions.

Suppose we have an asymptotic expansion of $f(n)$ : $$f(n)= a_1 \varphi_{1}(n)+a_2 \varphi_{2}(n)+O\big(\varphi_{3}(n)\big) \qquad (n \to \infty).$$

Dividing both parts by $ε > 0$ and taking the limit will produce $n → ∞$, the coefficient of the highest-order term in the expansion, which represents the rate at which $S_{n}$ changes in its leading term. $$\lim_{n\to\infty} \frac{f(n)}{\varphi_{1}(n)} = a_1.$$

Informally, one can say: "$φ_{1}(n)$ grows approximately as $a_{1}$". Taking the difference between $f(n)$ and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about $f(n)$: $$\lim_{n\to\infty} \frac{f(n)-a_1 \varphi_{1}(n)}{\varphi_{2}(n)} = a_2 .$$

Here one can say that the difference between the function and its approximation grows approximately as $a_{1}φ_{1}(n)$. The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself.

Informally, something along these lines happens when the sum, $S_{n}$, of independent identically distributed random variables, $f(n)$, is studied in classical probability theory. If each $X_{i}$ has finite mean $μ$, then by the law of large numbers, $f(n)$. If in addition each $X_{i}$ has finite variance $a_{2}φ_{2}(n)$, then by the central limit theorem, $$ \frac{S_n-n\mu}{\sqrt{n}} \to \xi ,$$ where $ξ$ is distributed as $X_{1}, ..., X_{n}$. This provides values of the first two constants in the informal expansion $$S_n \approx \mu n+\xi \sqrt{n}. $$

In the case where the $X_{i}$ do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors: $$\frac{S_n-a_n}{b_n} \rightarrow \Xi,$$ or informally $$S_n \approx a_n+\Xi b_n. $$

Distributions $S_{n}⁄n → μ$ which can arise in this way are called stable. Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor $b_{n}$ may be proportional to $n^{c}$, for any $σ^{2}$; it may also be multiplied by a slowly varying function of $n$.

The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function $N(0,σ^{2})$, intermediate in size between $n$ of the law of large numbers and $Ξ$ of the central limit theorem, provides a non-trivial limiting behavior.

Density functions
The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov for a particular local limit theorem for sums of independent and identically distributed random variables.

Characteristic functions
Since the characteristic function of a convolution is the product of the characteristic functions of the densities involved, the central limit theorem has yet another restatement: the product of the characteristic functions of a number of density functions becomes close to the characteristic function of the normal density as the number of density functions increases without bound, under the conditions stated above. Specifically, an appropriate scaling factor needs to be applied to the argument of the characteristic function.

An equivalent statement can be made about Fourier transforms, since the characteristic function is essentially a Fourier transform.

Calculating the variance
Let $S_{n}$ be the sum of $n$ random variables. Many central limit theorems provide conditions such that $c ≥ 1⁄2$ converges in distribution to $√n log log n$ (the normal distribution with mean 0, variance 1) as $√n$. In some cases, it is possible to find a constant $S_{n}/√Var(S_{n})$ and function $f(n)$ such that $N(0,1)$ converges in distribution to $n → ∞$ as $σ^{2}$.

$$

Products of positive random variables
The logarithm of a product is simply the sum of the logarithms of the factors. Therefore, when the logarithm of a product of random variables that take only positive values approaches a normal distribution, the product itself approaches a log-normal distribution. Many physical quantities (especially mass or length, which are a matter of scale and cannot be negative) are the products of different random factors, so they follow a log-normal distribution. This multiplicative version of the central limit theorem is sometimes called Gibrat's law.

Whereas the central limit theorem for sums of random variables requires the condition of finite variance, the corresponding theorem for products requires the corresponding condition that the density function be square-integrable.

Beyond the classical framework
Asymptotic normality, that is, convergence to the normal distribution after appropriate shift and rescaling, is a phenomenon much more general than the classical framework treated above, namely, sums of independent random variables (or vectors). New frameworks are revealed from time to time; no single unifying framework is available for now.

Convex body
$$

These two $ε_{n}$-close distributions have densities (in fact, log-concave densities), thus, the total variance distance between them is the integral of the absolute value of the difference between the densities. Convergence in total variation is stronger than weak convergence.

An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".

Another example: $S_{n}/(σ√n⋅f(n))$ where $N(0,1)$ and $n→ ∞$. If $ε_{n} ↓ 0$ then $n ≥ 1$ factorizes into $X_{1}, ..., X_{n}$ which means $f(x_{1}, ..., x_{n}) = f(|x_{1}|, ..., |x_{n}|)$ are independent. In general, however, they are dependent.

The condition $x_{1}, ..., x_{n}$ ensures that $E(X2 k) = 1$ are of zero mean and uncorrelated; still, they need not be independent, nor even pairwise independent. By the way, pairwise independence cannot replace independence in the classical central limit theorem.

Here is a Berry–Esseen type result.

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The distribution of $k = 1, ..., n$ need not be approximately normal (in fact, it can be uniform). However, the distribution of $f(x_{1}, ..., x_{n}) = const · exp(−(|x_{1}|^{α} + ⋯ + |x_{n}|^{α})^{β})$ is close to $ \mathcal{N}(0, 1)$ (in the total variation distance) for most vectors $α > 1$ according to the uniform distribution on the sphere $αβ > 1$.

Lacunary trigonometric series
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Gaussian polytopes
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The same also holds in all dimensions greater than 2.

The polytope $K_{n}$ is called a Gaussian random polytope.

A similar result holds for the number of vertices (of the Gaussian polytope), the number of edges, and in fact, faces of all dimensions.

Linear functions of orthogonal matrices
A linear function of a matrix $β = 1$ is a linear combination of its elements (with given coefficients), $f(x_{1}, ..., x_{n})$ where $const · exp (−|x_{1}|^{α}) … exp(−|x_{n}|^{α}),$ is the matrix of the coefficients; see Trace (linear algebra).

A random orthogonal matrix is said to be distributed uniformly, if its distribution is the normalized Haar measure on the orthogonal group $X_{1}, ..., X_{n}$; see Rotation matrix.

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Subsequences
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Random walk on a crystal lattice
The central limit theorem may be established for the simple random walk on a crystal lattice (an infinite-fold abelian covering graph over a finite graph), and is used for design of crystal structures.

Applications and examples
A simple example of the central limit theorem is rolling many identical, unbiased dice. The distribution of the sum (or average) of the rolled numbers will be well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution. It also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.

Regression
Regression analysis, and in particular ordinary least squares, specifies that a dependent variable depends according to some function upon one or more independent variables, with an additive error term. Various types of statistical inference on the regression assume that the error term is normally distributed. This assumption can be justified by assuming that the error term is actually the sum of many independent error terms; even if the individual error terms are not normally distributed, by the central limit theorem their sum can be well approximated by a normal distribution.

Other illustrations
Given its importance to statistics, a number of papers and computer packages are available that demonstrate the convergence involved in the central limit theorem.

History
Dutch mathematician Henk Tijms writes:

"The central limit theorem has an interesting history. The first version of this theorem was postulated by the French-born mathematician Abraham de Moivre who, in a remarkable article published in 1733, used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This finding was far ahead of its time, and was nearly forgotten until the famous French mathematician Pierre-Simon Laplace rescued it from obscurity in his monumental work Théorie analytique des probabilités, which was published in 1812. Laplace expanded De Moivre's finding by approximating the binomial distribution with the normal distribution. But as with De Moivre, Laplace's finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematician Aleksandr Lyapunov defined it in general terms and proved precisely how it worked mathematically. Nowadays, the central limit theorem is considered to be the unofficial sovereign of probability theory."

Sir Francis Galton described the Central Limit Theorem in this way:

"I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the 'Law of Frequency of Error'. The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshalled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along."

The actual term "central limit theorem" (in German: "zentraler Grenzwertsatz") was first used by George Pólya in 1920 in the title of a paper. Pólya referred to the theorem as "central" due to its importance in probability theory. According to Le Cam, the French school of probability interprets the word central in the sense that "it describes the behaviour of the centre of the distribution as opposed to its tails". The abstract of the paper On the central limit theorem of calculus of probability and the problem of moments by Pólya in 1920 translates as follows.

"The occurrence of the Gaussian probability density $f(x_{1}, ..., x_{n}) = f(|x_{1}|, ..., |x_{n}|)$ in repeated experiments, in errors of measurements, which result in the combination of very many and very small elementary errors, in diffusion processes etc., can be explained, as is well-known, by the very same limit theorem, which plays a central role in the calculus of probability. The actual discoverer of this limit theorem is to be named Laplace; it is likely that its rigorous proof was first given by Tschebyscheff and its sharpest formulation can be found, as far as I am aware of, in an article by Liapounoff. ..."

A thorough account of the theorem's history, detailing Laplace's foundational work, as well as Cauchy's, Bessel's and Poisson's contributions, is provided by Hald. Two historical accounts, one covering the development from Laplace to Cauchy, the second the contributions by von Mises, Pólya, Lindeberg, Lévy, and Cramér during the 1920s, are given by Hans Fischer. Le Cam describes a period around 1935. Bernstein presents a historical discussion focusing on the work of Pafnuty Chebyshev and his students Andrey Markov and Aleksandr Lyapunov that led to the first proofs of the CLT in a general setting.

A curious footnote to the history of the Central Limit Theorem is that a proof of a result similar to the 1922 Lindeberg CLT was the subject of Alan Turing's 1934 Fellowship Dissertation for King's College at the University of Cambridge. Only after submitting the work did Turing learn it had already been proved. Consequently, Turing's dissertation was not published.

title: Central limit theorem