Talk:Central limit theorem

message: Just a note: o(t2), t → 0, refers to a function which goes to zero more quickly than t2 (like t3), and not a function 'like' t2, which would be O(t2).  Hence, I have reverted the recent edits that changed o(t2) to o(t3).  Notably, the article on Big-O notation does not discuss limits other than the limit as t → ∞.  However, it should do so!  Ben Cairns 06:56, 14 Feb 2005 (UTC)
thread_title:  ''o''(''t''<sup>2</sup>) 

message: Sorry, I 've did not seen your message (Bjcairns) in the discussion enrty. I confused big O with small o. I though
that this o is reffering to the higher order corrections of the Taylor's expansion formula. I suppose that you
are right so I changed the article back to its previous version with o(t2) without being logged in.
That ip 143.233.xxx.xxx etc is mine :) My version is perhaps correct if we consider the Big O and not the small one. 
Theofilatos 17:07, 17 Feb 2005 (UTC)
thread_title:  ''o''(''t''<sup>2</sup>) Reply 

message: This article seems to be very mathematically complex. It could benefit from some simple layman's language. Ian Howlett 13:24, 30 June 2005 (UTC)
thread_title:  Needs layman's language too 

message: Quotation marks around a word often mean something like: that's what some people are often heard to call it, but I don't want to commit myself to agreeing.  Thus they de-emphasize.  If you write "John has a 'degree' from the University of the Ozarks", the quotation marks enclosing the word "degree" mean that maybe John and some others call it a "degree", but you don't necessarily agree.  Often quotation marks mean "don't take this word literally."  That is the meaning of the quotation marks around "the" in the section heading that says "The" central limit theorem.  The word "the" in this context implies uniqueness: that there is only one central limit theorem.  In fact there are many, with varying assumptions: sometimes independence is relaxed; sometimes identical distribution is relaxed; sometimes the random variables live in some space besides the real line, etc.  The quotation marks mean that often people call this one "the" central limit theorem, but the word "the" should not be taken too literally. Michael Hardy 18:16, 16 September 2005 (UTC)
replies: text: I am aware of the use of quotes in this way, I use them like that "every" day. :) However, I find it strange that somebody would quote the word the. Oleg Alexandrov 18:42, 16 September 2005 (UTC)
replies: text:  Ironic emphasis of "the" is common enough in informal American English (dunno if the Brits use it too). I don't think we want an easily misunderstood wordplay here. I've replace "The" central limit theorem with Classical central limit theorem. Feel free to find a different adjective. There are other uses of "scare quotes" in the article which should be reviewed. Regards & happy editing, Wile E. Heresiarch 03:08, 19 September 2005 (UTC)
thread_title: Quotation marks de-emphasize

message: Hi guys I'm an editor who delves a lot in physics (statistical mechanics especially) and a bit in statistics. That means I use this theorem a lot. I'll have other things to say, but as of now I just need to share something that sprung to my mind (not that it's original work, someone probably thought of that before me):
there is very probably a link between the non-independant case and polymer physics. A real-world polymer is basically a correlated random walk, although this correlation tends to decrease exponentially. Yet the object follows the Central Limit Theorem. About this see the ideal chain and worm-like chain articles, especially the parts about the Kuhn segment.
Either mathematicians have a version of non-independant CLT corresponding to this, in which case as a polymer and random walk editor I need to know, or this should probably be added as another case of non-independant CLT, in some form or the other.(ThorinMuglindir 23:40, 25 October 2005 (UTC)
thread_title: Link to polymers

message: As I said yesterday I am very much into editing polymer and random walk stuff for physics, which leads me to linking to CLT a lot. There is a case where convergence toward CLT is singular, yet arises quite often in random walks (namely, that is random walk on a lattice).
Take for instance independant variables which can take -1 or 1 for values with proba 1/2 each, and sum them N times.
If you look at the density function you obtain, it is not strictly a gaussian. It is a series of successive dirac delta functions. Now CLT is not that far off because the amplitude of the Dirac peaks of the resulting sum varies according to a gaussian curve. So that if you look at the function in the low fourier modes, it will correspond to the gaussian curve that is predicted by CLT. For high fourier modes (k > N.2Π, or k > 2Π if you don't renormalize the sum) the density of the resulting sum has nothing to do with a gaussian.
The situation is not the same if you consider a sum of continuous variables, or a lattice-free random walk, the same problem does not arise.
For example consider the countinuous variable that is uniformely distributed in [-1/\sqrt2;1/\sqrt 2], and sum a large number of independant realisations of this variable. This new variable has the same mean and variance as the previous one, yet you won't obtain a series of dirac peaks like in the previous case. The resulting density density will look as a gaussian pretty much at any scale, including in the high fourier range...
All this will probably be clearer by writing a formula: for the latter variable (variance 1, mean 0), the (normalized) sum converges toward the density function P(X), corresponding to N(0,1/N). Now, strictly speaking, the former variable sum's density function does not converge toward P(X), but rather toward:
:P'(X) = P(X)\Sigma_{k=-N}^{+N}\delta(X-k/N), where delta is the Dirac delta function
 
About this here are my questions: is the above somehow related to what you say about the nature of the third moment of the variable controlling the speed of convergence? Can this difference in convergence in the high and low fourier modes be formalized mathematically?(192.54.193.37 08:58, 26 October 2005 (UTC)
replies: text: Not all of what you are saying is immediately clear to me, but you seem to be talking about discrete random variables compared with continuous random variables.  Discrete random variables do not have probability density functions, but the cnetral limit theorem is not about densities anyway.  It is about convergence in distribution, i.e. about cumulative distribution functions.  So there is less problem about comparing discrete and continuous random variables.  The classic example is the normal approximation to the binomial distribution; even here the approximation can be misleading in the tails, as it often is when appling the central limit theorem. --Henrygb 23:55, 26 October 2005 (UTC)
replies: text: thanks my question was indeed related to that binomial distribution. Just as a remark it is often possible and useful to define a probability density function for a discrete variable, using Dirac delta function. Mathematically speaking Dirac delta function is not a function, but it's still a distribution (distribution, not in the sense of statistics, but in the sense of topology, that is to say, an object in the adherence of the space of functions). Of coures when you do physics you couldn't care less about what exactly is a function and what is a distribution... What I wrote above is just a reformulation of the meaning of the graph that compares the curve and the histogram in the binomial distribution article, reformulation that is based on Dirac delta functions.(ThorinMuglindir 10:03, 27 October 2005 (UTC)

text: if we build a histogram of the realisations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the histogram converges toward a Gaussian curve as n approaches infinity
The above is wrong. Counter example: a discrete variable that takes only even values. It's wrong to talk about "centers of the upper faces of the rectangles", one should tak about some  smoothed histogram, or much better (though less intuitive) about distribution functions.  — Preceding unsigned comment added by Leonbloy (talk • contribs) 10:40, 15 February 2011 (UTC)

text: I don't see it's necessary to talk about 'some smoothed histogram'—one can simply use a histogram with a larger bin width, e.g. 2 for your example. Qwfp (talk) 11:26, 15 February 2011 (UTC)
thread_title: Some singular cases that might be worth explaining

message: The first paragraph states: The most important and famous result is called simply The Central Limit Theorem which states that if the sum of the variables has a finite variance, then it will be approximately normally distributed. 
The random variables must have finite variance, right? This was the impression I got from http://mathworld.wolfram.com/CentralLimitTheorem.html. I am not skilled at mathematics so I do not know if saying the sum has a finite variance is correct. Thank you. Jason Katz-Brown 05:32, 11 February 2006 (UTC)
replies: text: They say the same thing: variances are non-negative so the sum of a finite number of them will be finite if and only if each of them is. --Henrygb 16:31, 11 February 2006 (UTC)
thread_title:  sum has finite variance, or the random numbers themselves? 

message: From the article:
:This means that if we build an organigram of the realisations of the sum of n independent identical discrete variables, the curve that joins the centers of the upper faces of the rectangles forming the organigram converges toward a gaussian curve as n approaches \infty. 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.
Huh? We draw an organizational chart of what, and how? I suppose "independent identical" is meant to be iid (as independent and identical is a contradiction), but what about the rest of it? Given the binomial distribution article reference, "organigram" is probably meant to be "histogram", though I don't see how the curve would join the "centers" of the upper faces more than any other points on them (a description which makes more sense for organigrams, even though as a whole using organigrams to depict distributions would be a strange idea). The histogram, then, is presumably of the probability distribution of a random variable that is the sum of n iid discrete random variables (or approximation of the same by frequencies in a finite sample of the random variable, but then we need to take limit at infinite sample size or the other limit won't converge). Is this interpretation correct? I'm not sure. How to explain it in good encyclopedia style? I don't know. As it stands this part of the article is very confusing and should be fixed, preferably by someone who knows something about probability theory (I don't, so I'm not touching it). 82.103.195.147 20:51, 12 August 2006 (UTC)
thread_title:  Organigram?! 

message: The need for a large sample size should be included. n >= 30 to 70 for it to be large. 70.111.238.17 14:11, 1 October 2006 (UTC)
replies: text: It says as n approaches ∞, and that is certainly quite large.  But some "rules of thumb" could be added too.  In many cases, "≥ 30" is quite conservative. Michael Hardy 21:29, 2 October 2006 (UTC)
thread_title:  Large sample size 

message: The article now says:
:: This is a limit theorem and is about central limits.
I've long thought it was the central theorem on limits, not the theorem on central limits.  Can someone explain just what "central limits" are?  The article's present comment seems to confuse rather than to clarify. Michael Hardy 02:51, 6 November 2006 (UTC)
replies: text: I removed a bit that said that the theorem was NOT a 'central' theorem, but was a theorem about 'central limits'. The text now says what it is, not what it isn't. I hope that clears it up. 8_)--Light current 03:24, 6 November 2006 (UTC)

text: I dont think you are correct. You should read the whole thing then youll see. Excerpt from page:
Note the following apparent "paradox": by adding many independent identically distributed positive variables, one gets approximately a normal distribution. But for every normally distributed variable, the probability that it is negative is non-zero! How is it possible to get negative numbers from adding only positives? The reason is simple: the theorem applies to terms centered about the mean. Without that standardization, the distribution would, as intuition suggests, escape away to infinity.
My itals, bolding--Light current 03:52, 6 November 2006 (UTC)

text: Ive removed the controversial statement until we can get the proper dope on it 8-)--Light current 04:01, 6 November 2006 (UTC)

text: I just (two years later!) added this ref to the history section of the article. Qwfp (talk) 14:58, 15 May 2009 (UTC)
replies: text: Why don't you simply write "central limit-value-theorem" or "central limit-theorem"? Chezistari (talk) 11:12, 11 January 2011 (UTC)
thread_title: central {limit theorem} or {central limit} theorem?

message: * I reckon the section on Classical CLT should start by stating the theorem. Justification should come after this. So the passage might read: " The central limit theorem says that the means of samples are normally distributed." - comments please —The preceding unsigned comment was added by 212.159.75.167 (talk) 20:18, 3 January 2007 (UTC)
thread_title:  Making it Easier to Understand 

message: I have heard that 30 individuals samples will meet the requirements of the C.L.T and therefore be considered a "statically" valid sample, assuming they were randomly selected using proper statistical procedure. This seems wrong to me, but I'd like to know what others say. Gautam  06:45, 8 June 2007 (UTC)
replies: text: Often far less than 30 is enough; sometimes many more is not.  It depends on what distribution you're sampling from.  But don't call them samples; call them observations within a sample. Michael Hardy 06:48, 8 June 2007 (UTC)
thread_title:  30 Individual Samples 

message: The CLT indicates for large sample size (n>29 or 100),[1] that the sampling distribution will have the same mean as the population, but variance divided by sample size
The CLT doesn't say that, and it doesn't depend on large sample size.  The expected value of the sum is the sum of the expected values, always.  For independent random variables the variance of the sum is always the sum of the variances. 72.75.76.121 (talk) 23:43, 5 December 2007 (UTC)
thread_title:  Additive mean and variance 

message: In the following:
Consider the sum Sn = X1 + ... + Xn.
Then the expected value of Sn is nμ and its standard error is σ n−1/2. Furthermore, informally speaking, the distribution of Sn approaches the normal distribution N(nμ,σ2n) as n approaches ∞.  
σ n−1/2 should be σ n1/2, as this refers to the standard error of the sum, Sn, not of the sample mean. I made this change.tom fisher-york (talk) 16:30, 14 December 2007 (UTC)
thread_title:  Standard error of sum is σ&nbsp;''n''<sup>1/2</sup> 

message: It would be great if this article included a bit about why the CLT is so cool..
For those of us who are getting into some real math through wikipedia, it's really great when there is an example application of a theorem like CLT.
For people who already know all about CLT, it may seem like an impossible task to select an example (like giving an example use of addition)
but I think the CLT is just arcane enough to warrant a bit of practical stuff so people can see why it is so cool. For me, I was just trying 
to figure out what the probability of something -- something with a distinctly -not-normal distribution. I ran 1,000 simulations and
saw (naturally) that the simulations were generating a distribution with a very familiar (normal) distribution. Wow, I thought. That's
the Central Limit Theorem at work! So I was able to avoid further simulations and use a plain old Z test to estimate the final result -- p < e-26
. Even with a very fast computer, you could not do enough simulations of my problem to establish this probability. With the CLT, there is no need.
Just my 2 cents. I could add this myself if folks agree...
Tombadog (talk) 12:45, 13 March 2008 (UTC)
thread_title:  Uses of CLT? 

message: Do the random variables really have to be iid? Does the CLT work if the variables are independent but have different distributions? Let's say a build a house in 10 (or more) following steps from to basis to roof. Every step takes some time (lenght=random variable). The steps are independent from each other but have different distributions (normal, exponentiell, equal, whatever..). The total length (sum of the 10 random variables) should then be approximately normal distributed regardless of the distribution of each single variable (or is it not)? --217.83.60.191 (talk) 16:25, 18 March 2008 (UTC)
replies: text: If you simply drop the assumption of identical distribution, then the resulting statement is not true.  But the assumption of identical distribution can be replaced by any of various other assumptions with the result that the theorem is still true.  I don't have any of the details at the tip of my tongue, but probably such things should be added to the article. Michael Hardy (talk) 17:02, 18 March 2008 (UTC)

text: See discussion above under "What about the sum of non-identically distributed random variables". Melcombe (talk) 15:07, 2 April 2008 (UTC)
thread_title:  iid? 

message: I remember back in the old days at university when I heard of an eplanation of the CLT in real life by a friend of a friend. Most things in nature a normally distributed like the length of some kind of plant, the lifespan of some kind of animal, the temperature on a special day or whatever. All these random variables are influenced by many other variables. So the normal distribution is so important because everything (every variable) that is the result of many many other variables is approx. normally distributed. It is not really the sum of other variables in a mathematical way as we don't know exactely the relationship between the variables but its close to that. --Unify (talk) 23:46, 18 March 2008 (UTC)
thread_title:  CLT in real life? 

message: Under "Proof of the central limit theorem", the artlicle has
::For any random variable, Y, with zero mean and unit variance (var(Y) = 1), the characteristic function of Y is, by Taylor's theorem,
:::\varphi_Y(t) = 1 - {t^2 \over 2} + o(t^2), \quad t \rightarrow 0
The correction required relates to "by Taylor's theorem" ... the expansion might be a Taylor expansion, but the result that the expansion is valid in this case is not (i.e. assuming that the variance exists and not assuming higher order moments). Anyone have a proper reference for the result, or should the sentence just be rephrased?
Melcombe (talk) 15:16, 2 April 2008 (UTC)
thread_title: Correction needed

message: I think the statement about the "remarkably simple proof" is not appropriate. The version of Taylor's theorem that is used here, is probably known only to a small fraction of the readers (\phi_Y(t) has complex values), the "simple properties of characteristic functions" (referring to linear transformations) are not explained, and the convergence to an exponential would deserve a reference. Above all, the heavy tool applied here, the Levy continuity theorem, is not trivial at all. In my opinion, expressing a proof in terms of another difficult theorem doesn't make it simple. It would therefore be more appropriate to state that rigorous proofs of the central limit theorem tend to be cumbersome, but that a non-rigourous argument is easily obtained from the Taylor expansion of the characteristic functions.  —Preceding unsigned comment added by 84.56.2.132 (talk) 06:59, 17 July 2008 (UTC)
replies: text: The “o” symbol here refers to the 1/n part, and refers to the n → ∞ limit. Thus you could replace o(t2/n) by t2/n3/2, but not by t3/n. As for the Lévy theorem, it is certainly both crucial for the proof and nontrivial. However that theorem is much more intuitive—that is easier to believe in—than CLT. If a sequence of characteristic functions converges, then so must the random variables too — might be difficult to prove, but the statement seems so obvious that it seems possible for the elementary level proof to skip it.  // stpasha »  22:01, 30 May 2010 (UTC)
thread_title: Proof of the central limit theorem

message: I add a ref about CLT for convex bodies, and remove the tag. Unfortunately, the tag "expert attention needed" was inserted with no explanation, what for. Thus, I am not sure: is it still needed, or not? Boris Tsirelson (talk) 19:52, 18 October 2008 (UTC)
replies: text: Well, other than adding reference(s) to this section, nothing much has changed. While I didn't add the tag, there seem to be two points needing attention. Firstly, for the first three items in the list, it would be good to have an outline of the sort of conditions/assumptions being imposed for the CLT to hold: possibly a single summary might do for all three? Secondly, for the 4'th item (convex bodies) it seems necessary to say how a CLT can apply to "bodies" or "sets" when all the discussion above is about random things which are numerically-valued. I guess there would be the question of whether such cases should be dealt with under the heading of "dependence", or might be better with their own slot, depending on what is being meant. Melcombe (talk) 14:15, 22 October 2008 (UTC)

text: There is also the question of the heading "lack of independence". At least for some of the topics indicated, it seems that both "non-identical" and "non-independent" are allowed, whereas there is at least an implication that what is covered is the case of "identical" and "non-independent". Melcombe (talk) 14:27, 22 October 2008 (UTC)
replies: text: I see. Well, maybe I'll do some more. Why "lack of independence"? For two reasons. First, some time ago the main part of the article covered the non-identical case. Second (and more important), "non-identical" is a relatively small problem, while "non-independent" is relatively hard. About convex bodies: it is meant that a point is chosen from a given convex body (uniformly); its coordinates are "random things which are numerically-valued", but dependent in a way not typical for probability theory (till now); to cover this case is considered an important progress. Boris Tsirelson (talk) 15:21, 22 October 2008 (UTC)
replies: text: OK, some improvements would be good. Regarding "convex bodies", it is not obvious how this might differ from a non-independent version of what is in the subsection titled "Multidimensional central limit theorem" ... what role does the "convex" bit have? Is it just to ensure that the mean is within the "body"? The abstract and first page of the cited article are not particularly informative to me. Melcombe (talk) 09:12, 23 October 2008 (UTC)

text: I am reading some sources. About convex body: it is completely different, since the large parameter is not the number of summands (this is just 1, - no sum at all, just a single random vector) but rather the dimension of the body (and the random vector), that is, the number of (random, dependent) coordinates. But wait, I'll try to write it down some day. Boris Tsirelson (talk) 13:53, 23 October 2008 (UTC)

text: I did something, and shall continue. Boris Tsirelson (talk) 20:15, 23 October 2008 (UTC)
thread_title: Lack of independence: "Expert attention"

message: I suspect the Lindeberg condition in the section discussing non-identically distributed random variables might be incorrect. Possibly there should be an average instead of a plain sum, or someting like that. I suspect that in the current form, the condition is essentially never satisfied. Not sure about this, though. I'll check this if I find the original reference. --130.231.89.82 (talk) 09:19, 22 October 2008 (UTC)
replies: text: Why incorrect? The sum in it corresponds to the sum in the definition of  s_n.  It means that deviated values do not contribute to the variance (in the limit). I believe it is correct. Boris Tsirelson (talk) 20:21, 23 October 2008 (UTC)
thread_title: Error in "Lindeberg condition"?

message: In this new (sub)section, there is "for all t in R" ... but t only appears as a dummy integration variable. Melcombe (talk) 16:26, 28 October 2008 (UTC)
replies: text: Oops... You are right, thank you. I'll correct it soon. In fact, I see that the formulation is clumsy; we can just say "converge in distribution". (In contrast, "CLT for convex bodies" includes some uniformity over all bodies or densities.) Strangely, the clumsy formulation is used by the authors. Boris Tsirelson (talk) 18:55, 28 October 2008 (UTC)
thread_title:  Error in Central limit theorem for Gaussian polytopes 

message: To User:Melcombe (and maybe someone else): Probably you could add some results about asymptotic normality for statistical estimators. Boris Tsirelson (talk) 07:08, 4 November 2008 (UTC)
thread_title:  Asymptotic normality for statistical estimators 

message: : Surely the reference to Turing's fellowship disertation should refer not to Cambridge University but to King's College Cambridge! 86.22.72.56 (talk) 18:13, 10 November 2009 (UTC)
thread_title:  Turing and the CLT 

message: Can we be careful to define "sum of distribution functions".  Clearly this is not the process say SUM(U,V) = (U + V)/2 which indeed yields a distribution function with all the nice properties, and looks like a "sum".  The usage of convolution needs to be well explained and tied to the notion of the "distribution of the means".  If we have a random process which, say, is constrained outputs values in the range [0,1] then clearly the means will never reach a limit which is a normal distribution.
    Z_n = \frac{S_n - n \mu}{\sigma \sqrt{n}}\,, 
the above appears on the page  missing the forward slash for  division between sigma and sqrt(n)  —Preceding unsigned comment added by 76.76.233.148 (talk) 23:31, 14 November 2009 (UTC)
thread_title: Help, clarification

message: This appears to be an excellent article, particularly for those intelligent and informed enough to understand it or who are already aware of the subjects import. However the Lede fails to impart that it is a fact (since some equate "theorem" with "theory" and hence something that might be only conditionally true), it's relation to the quotidian use of statistical sampling and inference, or the fact that it's one of the most important results of mathematics. 72.228.177.92 (talk) 20:01, 12 February 2010 (UTC)
replies: text: It also gives the impression it's a recent result. 72.228.177.92 (talk) 16:52, 13 February 2010 (UTC)
thread_title:  Lede Shortcomings 

message: Can I prompt a look at Subindependence and its references? Those references imply that the CLT can be proved using subindependence rather than independence and, if so, it may be worth fitting this into the article at some appropriate place. Melcombe (talk) 10:25, 2 June 2010 (UTC)
replies: text: Rather a curiosity. With some effort it is possible to construct examples of subindependence (in the lack of independence). But I doubt that it is useful at least once. Boris Tsirelson (talk) 14:50, 2 June 2010 (UTC)
replies: text: But why does it have to be useful? Someone looking for good information about the CLT might either expect, or be happy to find, such information. After all, a lot of effort in maths is placed into finding weaker/weakest conditions where a known result holds. And I would say "I doubt that it is useful at least once" about many of the "extensions" already noted in the article. Still, unless someone else thinks otherwise, I won't push harder for its inclusion. Melcombe (talk) 09:34, 4 June 2010 (UTC)
thread_title:  Another "beyond" 

message: Is the diagram in the introduction about tossing one fair coin really an example of the CLT? There is only one random variable here, the outcome of the coin toss. It looks more an illustration that the binomial distribution tends to become the normal distribution for large n.
Additionally, the diagram is confusing because it looks like it is an illustration of the sentence in the introduction about how the distributions of the outcomes of a large number of UNFAIR coins (not just one coin) taken together become normally distributed. By the way, I found that a very helpful example.  —Preceding unsigned comment added by 86.85.210.34 (talk) 19:04, 18 January 2011 (UTC)
replies: text: If a coin is tossed twice, we get two (independent) random variables (and their sum, if we add them). --Boris Tsirelson (talk) 06:08, 19 January 2011 (UTC)

text: See de Moivre–Laplace theorem : "In probability theory, the de Moivre–Laplace theorem is a normal approximation to the binomial distribution. It is a special case of the central limit theorem." --Qwfp (talk) 10:42, 19 January 2011 (UTC)
thread_title:  Diagram in introduction 

message: For the classical CLT, the article currently says, "Suppose we are interested in the behavior of the sample average of these random variables [blah]" in the assumptions.  I bet the result also holds if we're not interested in the sample average at all.  I think this should be fixed.  How about "Let S_n = ... be the sample average..." and then just go from there?  — Preceding unsigned comment added by 192.150.186.248 (talk) 18:02, 1 June 2011 (UTC)
thread_title:  Strange unnecessary theorem hypothesis 

message: Don't you think that the section "history of CLT" could be an article itself? There's so much to tell about it! I expanded the section "multivariate central limit theorem" myself, but I also think it could be a separate article, due to the natural difficulties involving random vectors rather than random variables. Luizabpr (talk) 15:26, 25 June 2011 (UTC)
thread_title: The article is too big already

message: Why does the string UNIQ79b599716edd6743-math-0000007B-QINU appears in the statement of the multivariate version of the CLT? Can anyone clear it? Can't edit the article.  — Preceding unsigned comment added by Reempe123 (talk • contribs) 01:58, 19 October 2011 (UTC)
replies: text: I do not see anything like that in the article (neither in wikitext nor in rendered text). Boris Tsirelson (talk) 07:02, 19 October 2011 (UTC)
replies: text: I was seeing something very similar (different numbers in the middle) just now. I checked the wikitext and removed an empty  pair, which seems to have fixed it. Qwfp (talk) 10:04, 19 October 2011 (UTC)
thread_title:  Rubbish in Multivariate CLT 

message: For such an incredibly important topic this is a pretty ugly and unwieldy article. There is talk of the effect of repeated adding but one should talk about how the distribution of a sum of two r.v.'s involves a convolution and it is the repeated re-convolution of densities that attenuates the non-Normal characteristics of every density with finite variance. There should be discussion of stable distributions and the fact that the normal is the only stable distribution with finite variance and that this is why it is the end point of the unlimited convolution of any set of distributions of finite variance with themselves. Simply put, the higher order features are progressively diminished in the repeated convolution operation. This is actually a very easy thing to understand and removes the feeling of "magic" from the operation of the CLT. It works for a reason and that is the reason.  If you limit yourself to finite variances, you end up with the normal. If you had other constraints, then you might end up with one of the other stable distributions, such as the Levy flight.
Also how about a little discussion of those pathological distributions that fail to satisfy the CLT? It's not hard to find examples, after all. Cauchy distribution?  — Preceding unsigned comment added by 69.142.244.49 (talk) 01:52, 28 March 2012 (UTC)
replies: text: WP:Be bold. Qwfp (talk) 15:29, 28 March 2012 (UTC)

text: But those (normalized) iid sums converge to a stable distribution. These are not really pathological, just heavy tailed, and have many important applications. Mathstat (talk) 00:08, 29 March 2012 (UTC)
thread_title:  Ugly Article 

message: The phrase "The condition  is replaced with much weaker conditions:    for " is removed by User:Jura05 with the explanation that "logconcave is needed anyway". Of course, logconcavity is required, but what of it? Now the reader does not know, what does "more general" mean in this case. As for me, it was better before this edit. Boris Tsirelson (talk) 08:10, 4 June 2012 (UTC)
thread_title:  "Convex body" - more general case 

message: Recently, User:Rphirsch replaced "mean" (in the lead) with "parameters of location". True, many natural statistical estimators are asymptotically normal (for large samples). In particular, location; but not just location. However, what is called "Central Limit Theorem"? Is it about the sum, or equivalently, the mean (as I always believed)? Or about location parameters? Or about arbitrary parameters? Or about all cases of asympototic normality? Boris Tsirelson (talk) 16:50, 19 March 2013 (UTC)
thread_title:  Why parameter of location? 

message: In Extensions to the theorem:Products of positive random variables
there is a ref 19, which is not supporting the phrase (at least the paper's text does not mention any square integrability. I'm not a specialist in the field so, can someone double-check it?Jorgecarleitao (talk) 12:10, 13 June 2013 (UTC)
replies: text: Ref 19 begins with a strange phrase (and by the way, it does mention square integrability): "It is well known that the products of independent, identically distributed (iid), positive, square integrable random variables (rv’s) are asymptotically lognormal. This fact is an immediate consequence of the classical central limit theorem (clt)." Square integrability of a random variable is more than enough for ensuring square integrability of the positive part of its logarithm; but it cannot ensure square integrability of the negative part of the logarithm (since the random variable may take small values too often). Probably the quoted phrase should not be taken too seriously; it is not the formulation of a result. Results of the paper are on a different matter. Not a relevant source. Boris Tsirelson (talk) 14:24, 13 June 2013 (UTC)
thread_title:  Ref 19 does not seem to support sentence 

message: With certain reservations in relation to Wikipedia it is possible to paraphrase a conclusion, conformable to the central limit theorem:
If the material of article of Wikipedia is the sum of many independent texts of participants, each of which makes a small contribution relatively the general result, 
at increase in number of participants objectivity and completeness of coverage of a resultant material aspires to the ideal.Vladimir Baykov (talk) 19:22, 21 July 2013 (UTC)
thread_title:  The central limit theorem and Wikipedia 

message: All the tutorials, e.g. Khan and lss academies, teach that the normal distribution will have the population mean and 1/n the population variance. Why does this article silence down these key facts? Why does not tell more than the distribution is Normal? --Javalenok (talk) 11:36, 17 November 2013 (UTC)
replies: text: From Central limit theorem#Classical CLT: "For large enough n, the distribution of Sn is close to the normal distribution with mean µ and variance ". Qwfp (talk) 11:52, 17 November 2013 (UTC)
thread_title:  normal distribution with a mean equal to m and a variance equal to s² / n 

message: It doesn't appear to me that this quote is dealing with the central limit theorem.  Without knowing this actual reference, Galton generally preferred the median to the mean which is why he wants his group of arranged in order so that the median can easily be identified.  — Preceding unsigned comment added by Jonnano (talk • contribs) 13:47, 23 November 2013 (UTC)
thread_title:  Galton's quote 

message: Edits of 37.100.122.230 of Jan 9 introduce a number of evident errors.
In particular: multiplies by sigma in order to get standard (instead of dividing by sigma). (Before the formulation was correct: convergence to N(0,\sigma^2).)
Random variables converge to the normal density. (Before the formulation was correct: convergence in distribution.)
Uniform convergence of cumulative probabilities to the normal density. (Before the formulation was correct: to the normal cumulative function.)
I revert it all. Boris Tsirelson (talk) 20:36, 9 January 2014 (UTC)
thread_title:  Recent incompetent edits 

message: the article contains a translation of a passage from a paper of George Polya,  which begins as follows:
"The occurrence of the Gaussian probability density 1 = e−x2 in repeated experiments, in errors of measurements, . . . "
But the mathematical expression 1 = e−x2 makes no sense in this context.  Instead, the Gaussian probability density (with mean μ = 0) is as follows:
          d(x)  =  (1/(sqrt(2π) σ)) e−x2/2σ2
for some positive standard deviation σ.  
In the simplest case, which is the "standard" case with μ = 0 and σ = 1, this becomes
          d(x)  =  (1/sqrt(2π)) e−x2/2
Daqu (talk) 20:16, 4 June 2015 (UTC)
replies: text: Indeed. But someone should take the source and look what is really written there. Boris Tsirelson (talk) 04:51, 5 June 2015 (UTC)
thread_title:  Mistranslation of Polya 

message: Someone added this section by just copying Stable distribution#A generalized central limit theorem, not even bothered by technical incompatibilities. I've corrected the refs as needed (but still, their style is different from others). I am not sure, however, whether it was a good idea to copy the whole section hereto, or not. Boris Tsirelson (talk) 14:57, 12 June 2015 (UTC)
thread_title:  Generalised theorem 

message: Understanding the article's very first sentence depends crucially on understanding the word iterates. I tried to find a Wikipedia article on what an iterate is, but I got redirected to Iteration, where the term is not mentioned. --217.226.68.242 (talk) 22:16, 1 June 2016 (UTC)
replies: text: Yes. I guess what is meant, but I never saw this term. I guess, it is used by statisticians; really? Boris Tsirelson (talk) 05:22, 2 June 2016 (UTC)
thread_title:  iterates 

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thread_title:  External links modified 

message: In the (first) theorem in Section "CLT under weak dependence" it may seem that Xn12 is an evident typo and should be Xn2. But no, it is not! The twelfth moment appears indeed in the theorem according to the source  (Billingsley). Moreover, the stronger version of the theorem (given in the same subsection) shows the price (in terms of αn) of using the moment 2 + δ (but not 2). Boris Tsirelson (talk) 11:49, 8 December 2016 (UTC)
thread_title:  Twelfth moment 

message: From the article:
:Let {X1, …, Xn} be a random sample of size n—that is, a sequence of independent and identically distributed (i.i.d.) random variables drawn from a distribution of expected value given by µ and finite variance given by σ2.
First, I am not a specialist in this field. But, first it says that {X1, …, Xn} is a random sample, and after that it says that every X is a random variable. But a random variable is something else, it is defined here: https://en.wikipedia.org/wiki/Random_variable.
And the same confusing appears at the beginning of the article. The definition seems good to me, but the example is related to computing the mean of a number of observations, and in the definition it says that "independent random variables are added". This is very confusing from my point of view. Even if the example is correct, an example which should help understand the definition should be provided. Costinb7 (talk) 17:00, 29 April 2019 (UTC)
replies: text: See also Talk:Random_variable#Confusion_between_"variables"_and_"functions". Boris Tsirelson (talk) 18:14, 6 May 2019 (UTC)
thread_title:  There is a mix-up here between an observation and a random variable