Talk:Characteristic function (probability theory)

4708:, as the characteristic function is the Fourier transform of the pdf. If you haven't, the most familiar physical example of a Fourier transform is the relationship between waveforms and frequency spectra, e.g. of sound. (In quantum mechanics wavefunctions for position & momentum, and also energy & time, are related by Fourier transforms, which links to Stpasha's comment about Heisenberg's uncertainty principle above). Clearly things are simpler to visualise if the characteristic function is real, which is the case if the pdf is symmetric about zero (that's in the 'Properties' section of this article). About the simplest such example to derive in probability theory is a Bernoulli distribution taking values –½ and +½ with probabilities 4128:"The argument of the characteristic function will always belong to the same space where random variable X takes values" (Sect. "Definition") — really? In principle, it belongs to the dual space. In practice, the given space usually is endowed with an inner product and so, may be treated as dual to itself. But not always; the distinction may be essential for random processes (because of infinite dimension). In fact, the distinction manifests itself already in finite dimension, if the given space is just an abstract linear space (with no preferred basis). 172: 151: 81: 71: 53: 3766:'s suggestion, we could consider a basis in L for the eigenspace associated to eigenvalue 1. If you want to describe all elements of this eigenspace that are positive and integrable then it depends on what you mean by describe (aka why is it not enough to say take everything in the eigenspace that is positive and integrable.) I suppose it could be useful to notice that we are looking for 4261:. Then the algebraic dual is"larger" than the original space, and they are not isomorphic. That would mean that the characteristic function has"more dimensions" than the distribution function. If we consider the continuous duals however, then by Riestz representation theorem this space is isomorphic to the domain of the random variable, and there isn’t much error in saying that 22: 4057:

this by removing Im(phi) from the list, but I never saved my edit. I previewed it, only to find out I'd messed up the code on all the other symbols for Re(phi) etc. Could someone who knows what they're doing with the code please remove Im(phi)? (Unless it actually does belong there, but for the life of me I cannot see how!)

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comment is stated. I was not sure if the restriction applies to the Levy's formula or to the re-statement that is amenable for numerical computation. Nevertheless, I felt it important to put it in the record. After all, it does get you to think why Gil-Peleaz is necessary if Levy is fine enough by itself. Best Regards,

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onto the functions that are their own Fourier transform. From the reasons above we are interesting in finding positive functions that are integrable with integral 1 (from being a probability measure), and are bounded (from being the Fourier transform of a probability measure). All such functions are

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The characteristic function is just the Fourier transform of the distribution. (Thus it is really about distribution rather than random variable itself.) It is well-defined always. At least if we treat a distribution as a probability measure on the Borel sigma-field (be it completed or not). No place

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similar to the CDF. I think we can say they are both two common ways of identifying variables - or a stronger measure theoretic statement can be made such as they are equivalent methods of defining random variables measurably almost everywhere. I myself am too far gone since studying this to suggest

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However, mgf does not bring anything new to the theory compared to cf. Surely mgf can be used to calculate moments, or to formulate a convergence theorem. But cf already does that -- we can calculate moments by differentiating cf and we have Lévy continuity theorem. On the other hand mgf has quite a

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Thanks for explanaions. Actually, I asked about the physical meaning of t. For example, I understand the meaning of p in, say, binomial distribution, but I cannot link t with something that is sensible. A cumulative distribution function or a probability mass function (probability density function)

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How about providing a brief explanation of terms, such as t in e^it? That would be great to improve understanding of the cf for those who have not taken advanced courses in math. "and t∈R is the argument of the characteristic function" is not fully comprehensible. I can only conclude that this is a

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looking for probability measures whose Fourier transform is itself, but the Fourier transform of a probability measures is a bounded function, and the measure has integral one, so in fact our probability measure must have a bounded pdf. This pdf, being bounded and in L must also be in L. Following

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I hope you'll agree integrability is handled by the fact that I said Schwartz. I should have said something about positivity, but to ensure that one simply needs to take a function whose Fourier transform is positive. For completeness of this discussion here are a few examples aside the gaussian:

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The characteristic function ( FX(x)) provides an alternative way for describing a random variable (X). Similarly to the cumulative distribution function and then explanation to what E in this formula is. This way every symbol used is explained in the description. I am new to wikipedia and I am not

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I am a bit confused by this last comment, we seem to have moved from showing there are a few any other then the Gaussian distribution to describing the space of all probability distributions with this property. This problem is a bit harder. Here are some random thoughts about that. First we are

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The idea is just that we save some words using the idea of "inverse fourier transform" instead of "conjugate of the transform", and also help to see the inverse transform as something as natural as the "direct" transform. No transform domain is intrinsically better! there is no reason to insist in

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The article mentions that you can calculate the characteristic function taking the conjugate of the Fourier transform of the pdf, but isn't this simply the inverse Fourier transform?... This should be made more clear. We tend to think about transforming using only the "direct" transform first, and

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as not everything in my head got as far as the keyboard. I was envisaging pairs of graphs of a few symmetric pdfs and their characteristic functions as i wrote it, and it's the use of such graphs to illustrate characteristic functions that i haven't seen in a textbook. On reflection i don't think

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The article says that, if phi is a characteristic function, then so is Im(phi). But one property all characteristic functions phi have is phi(0) = 1, so Im(phi)(o) = Im(phi (0)) = Im(1) = 0, which is not 1. So, surely, if phi is a characteristic function, then Im(phi) never is? I tried correcting

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Some of the old notation involving was wrong and I have taken it out. Many people do define mgf's as allowing complex arguments. The similarity in the formulae for cf's and mgf's means that mgf's have to be mentioned early on to prevent readers wondering why they are not mentioned. On there is no

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into cf" will arrive at analytic cfs, ... it arrives at cfs that may or may not be analytic over regions to be defined, as per Lukacs. Just because you like thing think of real 't' only is no reason for the article to give the impression that only that case is important. And considering that the

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Thanks for the reply. I don't have access to Gil-Peleaz's original paper, which appears in Biometrika and our university does not have subscription to it, sadly. So I could not verify exactly why or under what condition this is true, if at all. Also, there is a bit of a confusion in the way the

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Under "properties" the article states "the characteristic function of a symmetric random variable is real-valued and even". A similar remark is made in the caption for graph of the characteristic function of a uniform distribution. But is this really generally true, or true only if the random

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into cf, we will arrive at something called analytic characteristic function (see Lukacs, ch.7), which loses some of the nice properties of cf (for example existence), but gains some extra useful properties (ridge property, existence of ∞ number of moments, sometimes entire-ness). It is when we

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I suggest you see page 11 of the Lukacs reference. He cites a theorem of Cramer (1946) which he says ensures the existence of the cf for "every distribution function". The previous context makes it clear that this includes even dfs with singular components. See also page 20, 36 and 64 for other

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if the function is positive definite the characteristic function will be positive, since these functions are themselves under the Fourier transform, this turns into a condition for the function itself to be positive. What would really be nice is to have a condition on the coefficients of the

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Hi. Well you could define a function like the characteristic function that works as you specify. But this would not take advantage of the structure present in matrices. The characteristic function as specified in the article is the standard, which crops up all over the place, at least in

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is a positive definite function". I suggest that this doesn't hold, and there are results that indirectly prove it isn't ... but I don't know of a direct approach to showing this. Note that "positive definite function" is a non-straightforward math thing if you are not familiar with it.

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variable is symetric around zero. For example, a normal or Cauchy distribution with a non-zero mean is symetric, but it appears that the characteristic functions would retain complex values. But I admit that I am weak on complex math, so maybe those i's that seem to be there drop out.

6018: 2076:). Your original question was about characteristic functions; they always exist for any distribution with a Lebesgue measure due to the dominated convergence theorem. I cannot say for sure about distributions with more pathological measures, but I suspect it's true there too. 4143:

You may be right, I haven’t seen a book which defines the cf for generic random elements. This sentence is just my attempt to describe those several definitions that follow. In most cases of course the dual space is isomorphic to the the “primal” space, so that for example if

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Would someone like to write a brief explanation of the meaning of a characteristic function, for those of us without such strong statistics and mathematics backgrounds? I have no idea of its concept or application. An example that appeals to intuition would be great, thanks!

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characteristic functions. Unless I made some big mistake, I think I proved that not all real-valued random variables have a characteristic function (in fact, I guess I showed that most of them don't - the problem with pathologies is that usually pathologies are the rule).

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reason why they should not be mentioned more than once if there is an increase in technicality towards the end of the article. It is wrong to think that either cf's or mgf's are only defined for real-valued arguments and I have tried to introduce something about this.

4744:, but if you think it helps i might put it in the article when i have time. Have to say, though, that as the main use of characteristic functions is for mathematical proofs, i'm not sure gaining intuition about their behaviour and relationship to pdfs is really much 4812:

How about explaining what "E" is ? I think we shouldn't assume any universal meaning of any notation, especially when people come from various countries to this article and may not understand what "E" or "M" is. For example first sentence of introduction should be:

6251: 2629:(mgf) removed from the "definition" section. The concept of characteristic function can and should be defined without any reference to mgf; in fact such reference is only confusing, because unlike cf, mgf may not always exist, or may be unbounded, etc. 1136: 2990:

Remember that this is not a text book. It needs to be useful to many types of people and to include what is notable about the topic and related topics. the connection to mgf's is definitely that. It is wrong to say that "to plug complex-valued

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is in the dual space, then which dual is it — algebraic or continuous? Of course the two duals coincide in finite dimensions, but suppose we want to consider the cf of an infinite-dimensional random element taking values in a Hilbert space

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Yes it is clear. But can I ask if this formulation is used in practice anywhere? It seems that if T were replaced by T-transpose then, not only would the formula be directly comparable to the vector case, but both the following would apply

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I do agree that moment-generating functions are treated in many current textbooks, although the reasons for that are either historical (mgf was invented earlier than cf), or the authors are unwilling to go into the area of complex

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is much better in the sense it is much more systematic, and much more complete. I just thought this was an interesting way to look at it, and it perhaps generates some closed formulas that are not covered by Boris's argument.

3270: 861: 6755: 3359:, and many others. For the concerned reader I should mention why these examples and you sum the iterates of Fourier transforms is not necessarily the Gaussian, one easy way to see this it to look at the asymptotic as 6801:

Which I think is dubious at best, and probably wrong. The writing goes on to define a CDF with new and unnecessary notation (the indicator function, we can save time and eyeballs to link other wiki articles on this).

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consider analytic cf, the connection with mgf becomes well-defined. However analytic cfs are different from regular cfs, they deserve their own section, and should not be implied in the definition of regular cf.

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then transforming back again, but you can perfectly say that you can find the characteristic function taking the inverse Fourier transform, then come back again taking the direct Fourier transform...

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is not quite the same as its characteristic function, depending on how you define the characteristic function. The characteristic function defined here is missing the normalizing factor involving 2

978: 4924:"Inversion formulas", after the Levy theorem: "This formula is valid only for strictly positive random variables." — Really?? Hard to believe. What exactly is meant? (This is a recent insertion by 326: 222: 1948: 508: 5029: 4396:

is reserved for the indicator function; and by the way, non-probabilists often call it characteristic function of a set! It is impossible to reserve every symbol once and for all. But if you see

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Probably you're correct. I don't think there is a contradiction exactly. Just many authors assume their random variables have a density, if this is the case the statements would be the same.

6646: 6573: 1739: 3908:, so take your favorite one and apply the projection. We still can't categorize them exactly, but we know that doing this we hit all of them. Here are some properties of all such functions 4606:). The point being that the product of these two uncertainties can never be smaller than a certain positive constant. Ok, I successfully confused myself. The “speed” of a random variable??? 4022:

is out). On the other hand if the function is smooth it cannot decay too slowly either. I guess this above paragraph goes under the "what would this imply" part of your original question.

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expansion in this basis that guaranteed that resulting function would be positive, and integrable. Unfortunately nothing springs to mind that would tell me when this was the case.

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Says so in N.G. Shepard's (1991a), "A Framework for Inversion", pg. 521 in the paragraph discussing Levy's formula. He cites this as the reason for the need for Gil-Peleaz formula.

1343:{\displaystyle {\overline {F}}_{X}(y)-{\overline {F}}_{X}(x)=\lim _{\tau \to +\infty }{\frac {1}{2\pi }}\int _{-\tau }^{+\tau }{\frac {e^{-itx}-e^{-ity}}{it}}\,\varphi _{X}(t)\,dt.} 6320: 3564: 6678: 5816: 5699: 4631:

Another suggestion is to provide examples of derivation cf for, say, bernulli, binomial and normal distributions. Such examples are useful for quick learning of what cfs are.

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I have a bit of a different definition for the inversion theorem that seems to contradict the one in this article. From "probability and random processes" by Grimmet I have:

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You linked to density function but wrote distribution function; they are not equivalent. Every real valued RV has a distribution function; not all have density functions (

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is defined over a domain of real numbers, it cannot be evaluated at an imaginary point (at least not from the point of view of strict mathematics). Of course if function

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Hermite polynomial) is an eigenfunction of cf. transform. Although not an appropriate pdf, we can always add a normal distribution to it to ensure positiveness. //

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In the above-named section, what the dimensions of the matrix T in relation to those of X? Are the dimensions the obvious ones or are other sizes used occasionally?

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The specific characteristic function mentioned is "well-known" ... at least it is in the table of characteristic functions that I have just added as a reference.

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It's like the pdf is already the "transform", and then the characteristic function is the "original" function, that we could see as a signal to be transformed...

127: 5283:{\displaystyle \displaystyle \lim _{T\to \infty }(2\pi )^{-1}\int _{-T}^{T}{\frac {e^{-ita}-e^{-itb}}{it}}\varphi (t)\,dt=\mu (a,b)+{\frac {1}{2}}\mu (\{a,b\}).} 540: 3000:

it needs to be included in the main definition. Also, before you add even more unexplained maths in a prvate notation into the initial section, read WP:LEDE.

6877: 4749: 212: 3820: 3581: 3147:, ... Thus, you may add to the normal density the fourth Hermite polynomial (times the normal density) with a small coefficient of such sign that for large 2927:

Anyways, i believe that mgf is nothing more than a "related concept", and thus should be mentioned in the corresponding section at the end of the article.

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is about physics, and it gives both cumulative and density as the probability concept. I don't know if there are pathological real valued random variables

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The list could probably be extended. For example, I it probably follows from the uncertainty principle that the function also cannot decay too rapidly (

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isn't easy, or maybe not even possible, but as an ex-physicist myself i appreciate where you're coming from in seeking to gain intuition about what

3184: 741: 4339:"I haven’t seen a book which defines the cf for generic random elements" — here is one: V.I. Bogachev, "Gaussian measures" (AMS); see Sect. 2.2. 6683: 103: 6862: 3971:

because by Bochner's theorem a positive function is the Fourier transform of a measure if and only if it is positive definite and continuous.

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http://math.stackexchange.com/questions/287138/moment-generating-functions-characteristic-functions-of-x-y-factor-implies-x/287321#287321

3055:. It turns out there are many distributions whose characteristic functions are themselves except for a normalizing factor in this way. 2328:. Since expectation operator exists for any random variable, and the corresponding integral is absolutely integrable for real values of 517: 6013:{\displaystyle \mathbb {E} e^{i\langle (X,Y),(\xi ,\eta )\rangle }=\mathbb {E} e^{i\,X\cdot \xi }\cdot \mathbb {E} e^{i\,Y\cdot \eta }} 3421: 4856: 6577: 6811: 4833: 4721: 94: 58: 4066: 1834:

Interesting but this is not what a physicist would call a positive definite function. I will therefore add a note to the main page

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Well, I see what happens, after looking at the articles by Shepard and Gil-Pelaez. They do not doubt that Levy theorem holds for

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random variable. I'm not entirely sure how mgf for a complex-valued r.v. is supposed to be defined, but most likely the argument

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the formulas would be equivalent to the vector case if both X and T were strung out into vectors using the usual vec() operator.

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real number, a continuous quantity. Does it always changes from -infinity to +infinity? And what is the role of this argument?

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And it is wrong to think that cf's or mgf's are defined for unreal-valued arguments. In fact, if we try to plug complex-valued

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contain easily interpreted arguments, while the interpretation of t in cf is vague for me. Can you explain me this, please?

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I agree that the inverse Fourier transform is closer than the forward Fourier transform. But there is also the factor of 2

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seeing the pdf as the correlate of a signal to be transformed by the direct fourier transform, and not the inverse one...

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On the other hand, you could present your findings here on the talk page, given that you did not have enough information.

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Rather strange. The link to Shephard 1991a does not work (for me today, at least). And here is a quote from a textbook:

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would have been discussed. However I think it's possible to think of it as some sort of “velocity”, borrowing from the

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satisfies all the requested properties but it implies that the corresponding distribution has vanishing second moment

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This text appears in the Definition Section. .. I may be blind, but I don't see any parentheses in that section..

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I think the formula for calculating the n-th moment based on the characteristic function was not correct. It was:

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any better text on this point. But the most important thing is not to say they're "similar" whatever that means.

6246:{\displaystyle F_{X}(a)-F_{X}(a-0)=\lim _{T\to \infty }{\frac {1}{2T}}\int _{-T}^{+T}e^{-ita}\varphi _{X}(t)\,dt} 4492: 3947:

is uniformly continuous and tends to 0 as |x| → ∞, because it is the Fourier transform of an integrable function.

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to denote the c.f. and the standard normal pdf. Only need to decide which one is which (I personally vote for

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real-valued random variables. However, since every characteristic function is continuous, there are only

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In fact, neither of those are correct. I will correct both occurrences of the formula in the article to

6261: 4906: 4793:(–½, ½) follows immediately from the general case of the uniform already given in the Examples section. 3519: 2535:

Hi Melcome. I've added some material to the page; hope it is clear. Let us know if not. Best wishes,

2097: 39: 4716:= ½. Then (similar to Stpasha's derivation immediately above) the characteristic function is ½e + ½e = 3568:

Although it certainly cannot be expressed as a Hermite polynomial since they have integer coefficients.

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that it's not possible to compute its characteristic function - but let's try a counting argument. Let

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Sorry i finished writing previous comment in a hurry earlier, resulting in an "it" with an unintended

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is the only random variable whose characteristic function coincides with its pdf? Then it would be a

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for c.f., and besides it clashes with the chi-square distribution... As an alternative, we can use

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You linked to density function but wrote distribution function; they are not equivalent - Baccyak4H

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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on Knowledge. If you would like to participate, please visit the project page, where you can join

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can be designed that does not have a characteristic function... Something along the lines of the

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denotes the real line, which means it goes from −∞ to +∞. Now, the characteristic function is a

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I suppose we have that actually we might be in better shape than I first thought. We have that

1656: 70: 52: 6480: 3077:=E (because only then Fourier transform is unitary), and then fixed point distribution will be 2481:{\displaystyle \left|\int e^{itx}dF_{X}(x)\right|\leq \int |e^{itx}|dF_{X}(x)=\int dF_{X}(x)=1} 6549: 6441: 5502: 5322: 4883: 4705: 4683: 4610: 4523: 4459: 4375: 4273: 4235: 4176:. In any case it may be just the matter of point of view: whether you want to see the product 4098: 3728: 3094: 3037: 2981: 2934: 2861: 2682: 2635: 2493: 2162: 2059: 2021: 1785: 1001: 6524: 5575: 5534: 5300: 4933: 4676: 4405: 4344: 4330: 4133: 4113: 4077: 3924:

must be bounded by 1, because it is the Fourier transform of a probability measure (itself).

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but in this case I do not understand why the characteristic function for some Levy stable

5366: 5034: 4728:(give or take a normalization constant) with characteristic function / Fourier transform 4290:"which dual is it — algebraic or continuous?" — continuous (otherwise, non-measurable...) 5824: 5639: 3272:, then when you apply a Fourier transform to such a thing you get out what you started. 6830: 6826: 6766: 6325: 4798: 4757: 4023: 3892:{\displaystyle {\tfrac {1}{4}}(I+{\mathcal {F}}+{\mathcal {F}}^{2}+{\mathcal {F}}^{3})} 3776: 3704:{\displaystyle {\frac {(-1)^{n}}{2^{n}(2n-1)!!}}H_{2n}({\sqrt {2\pi }}x)e^{-\pi x^{2}}} 3381: 3273: 3056: 2831:

and hopefully get a meaningful result. But what if there is no simple formula? What if

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P.S. Thanks for the reference by Durrett. I found a free ebook, and its fantastic. :)

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What does this phrase mean?: “which is essentially a different change of parameter”.

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results on the relation between the cf and whether the df has a singular component.

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so this quantity is trivially zero. Is this supposed to reflect some kind of limit?

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is real-valued, so it must be and even because its Fourier transform is real valued.

5614: 4094: 3724: 3090: 3033: 2977: 2930: 2489: 2158: 2113: 2055: 2017: 1861: 990: 4704:"means" and i've pondered it myself a while back. It helps if you've come across 6798:

The characteristic function is similar to the cumulative distribution function,

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Alright, so Hermite polynomials didn't really work; however density of the form

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The case of cf of the stochastic process is trickier though. Instead of having

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must be positive, and have integral one since it defines a probability measure.

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Yep, my bad. This should use alternative definition of characteristic function

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is a point where you want to evaluate your function. If you want to find what

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Comparing the characteristic function to the cumulative distribution function

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for the characteristic function in some textbooks, then we can follow these.

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Is it really the Fourier transform, or only *almost* the Fourier transform?

3576:'s example some more, and am now fairly sure that any function of the form 4293:"by Riestz representation theorem..." — and what if the original space is 6761:

to be reckoned with. What if we say that the characteristic function is

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which could be used to give some background on where c.f.'s came from. --

5818:-valued random variables. Then the following statements are equivalent. 4223:

belonging to some strange class of functions integrable in product with

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of a function, so i guess it is considered to be a trivial concept...

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is a reserved symbol for the pdf of the standard normal distribution.

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anyone could claim that would be original research though. That sinc(

4720:(½t). Another nice one to look at (not quite so easy to derive) is a 4717: 6568:

Doesn't this mean that the characteristic function is more like the

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It means that one function is just a rescaled version of the other:

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This looks like a very useful statement, relating independence of

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random variables. The point is quite different. Levy theorem gives

3119:(and no continuous spectrum). The corresponding eigenfunctions are 3111:

The unitary operator of Fourier transform has four eigenvalues: 1,

3509:{\displaystyle (ax^{4}-{\tfrac {3}{2\pi }}ax^{2}+1)e^{-\pi x^{2}}} 2136:; in the first case, it's trivial, in the second, well-order both 6591:

to worry about. The inverse Fourier transform is something like

4196:) between two elements of the same space over the field of reals. 2858:

is defined graphically for example. It is difficult to interpret

3151:

the addition is positive. This way you get an example you seek.

2996:

major work on characteristic functions defines them for complex

6561:

involve an integral involving exp(-ixt) or exp(-2πixt), with a

6067:(in the univariate case this means a point of discontinuity of 4455:

being the standard normal pdf, since it is more similar to Φ).

2124:(the psychomath's favourite tool in creating monsters) and the 2632:

The reference itself is quite dubious as well. The expression

988:, while former is generally used in statistical sciences. ... 15: 5741:. Is there a reason for the absence? (If not, I can add it.) 3032:

of the Fourier transformation; wonder what this would imply.

6557:

But among the various definitions of the Fourier transform,

4359:

How about we denote the characteristic function with letter

3875: 3858: 3847: 3242: 3216: 3196: 4849:

What's T suppose to denote? How different is this from t?

4570:

I haven't seen a book where the physical interpretation of

4515:=5 and compute E. There isn't even an article for the word 2788:

is given by some simple formula then we can always plug in

6339:

Sure, this is (a usual, convenient notation for) a limit:

4736:). I've never seen this in a probability textbook so it's 3371:, for the third it of course depends on what you pick for 3302:

Yes, but we also want it to be positive (and integrable).

1653:

is positive definite, the only points where it is not are

6427:{\displaystyle F_{X}(a-0)=\lim _{b\to a,b<a}F_{X}(b).} 4108:

You are right. Thanks. I just added "around the origin".

2144:

using the bijection (probably there's no need to use the

4647:

takes on only two values: 0 and 1, with probabilities 1−

2924:, but then such interpretation turns into a tautology. 5884:{\displaystyle \forall \eta ,\xi \in \mathbb {R} ^{d}} 5472:{\displaystyle F(b)=\lim _{a\to -\infty }(F(b)-F(a)).} 3825: 3445: 1741:

can be accepted since they show the same behaviour at

414:{\displaystyle E(X^{n})=(-i)^{n}\varphi _{X}^{(n)}(0)} 6686: 6654: 6597: 6483: 6444: 6345: 6264: 6094: 5897: 5853: 5827: 5795: 5756: 5707: 5668: 5642: 5505: 5485: 5398: 5369: 5325: 5086: 5085: 5057: 5037: 4969: 3823: 3584: 3522: 3424: 3187: 2903: 2864: 2837: 2817: 2794: 2767: 2724: 2685: 2638: 2338: 2228: 1877: 1789: 1747: 1682: 1659: 1604: 1534: 1456: 1397: 1139: 1028: 973:{\displaystyle \varphi _{2}(t)=\varphi _{1}(-2\pi t)} 919: 744: 672: 543: 437: 339: 252: 6572:

Fourier transform of the density function (if any)?

3123:

times the normal density. Their eigenvalues are: 1,

2148:

to create this monster). So, by counting, there are

2000:

can be described by a characteristic function; I am

1856:

Historia Matematica has a nice little thread on the

1002: 321:{\displaystyle E(X^{n})=-i^{n}\varphi _{X}^{(n)}(0)} 183:, a collaborative effort to improve the coverage of 98:, a collaborative effort to improve the coverage of 5392:alone. Theoretically, this is not a problem, since 3958:

need not be differentiable, or decay faster then 1/

1943:{\displaystyle E(X^{n})=i^{n}\varphi _{X}^{(n)}(0)} 503:{\displaystyle E(X^{n})=i^{n}\varphi _{X}^{(n)}(0)} 6749: 6672: 6640: 6511: 6469: 6426: 6314: 6245: 6012: 5883: 5839: 5810: 5781: 5733: 5693: 5654: 5517: 5491: 5479:But numerically, it is rather impractical to send 5471: 5384: 5355: 5282: 5069: 5043: 5024:{\displaystyle \varphi (t)=\int e^{itx}\,\mu (dx)} 5023: 3891: 3703: 3558: 3508: 3264: 2916: 2889: 2850: 2823: 2803: 2780: 2754:{\displaystyle M_{X}:\mathbb {R} \to \mathbb {R} } 2753: 2710: 2663: 2480: 2320: 1942: 1810: 1753: 1733: 1668: 1645: 1575: 1504: 1438: 1342: 1119: 972: 855: 729: 657: 502: 413: 320: 132:This article has not yet received a rating on the 6794:In the current body of text, the writing states: 4817:comfortable editing such important article yet. 4439:. Now I don't know if any textbook actually uses 3177:Another way is to consider any Schwartz function 2078:I cleaned up the wikilinks in your original post. 1386:There is something strange as it is formulated. 1120:{\displaystyle {\overline {F}}(x)={\frac {1}{2}}} 6375: 6146: 5415: 5088: 4418:Luckily we have an unambiguous notation for the 2108:be any set that is dense in ; is it possible to 1199: 1080: 6641:{\displaystyle \int F(\nu )e^{2\pi \nu t}d\nu } 1734:{\displaystyle \varphi (t)=e^{-|ct|^{\alpha }}} 4901:"and the last formula in parentheses is valid" 4507:=0 in the formula: E. If you are looking for 2140:and , sample uniformly from and get back to 8: 5947: 5911: 5270: 5258: 4845:Definition for k×p-dimensional random matrix 1528:I believe the problem lies in condition: " 730:{\displaystyle \phi ^{(n)}(0)=i^{n}EX^{n};} 6548:, then the characteristic function is the 6051: 4752:, but i've not used them for that myself. 3983:must be supported on the entire real line. 3770:integrable functions in this space. By a 1782:to see what is required .. it is not just 984:Second definition is more standard in the 145: 47: 6732: 6714: 6687: 6685: 6653: 6617: 6596: 6539:The introduction contains this sentence: 6488: 6482: 6449: 6443: 6406: 6378: 6350: 6344: 6297: 6269: 6263: 6220: 6201: 6188: 6180: 6161: 6149: 6121: 6099: 6093: 5993: 5985: 5984: 5964: 5956: 5955: 5907: 5899: 5898: 5896: 5875: 5871: 5870: 5852: 5826: 5802: 5798: 5797: 5794: 5773: 5755: 5734:{\displaystyle \varphi _{X},\varphi _{Y}} 5725: 5712: 5706: 5673: 5667: 5641: 5504: 5484: 5418: 5397: 5368: 5324: 5242: 5172: 5150: 5143: 5137: 5129: 5116: 5091: 5084: 5056: 5036: 4992: 4968: 3880: 3874: 3873: 3863: 3857: 3856: 3846: 3845: 3824: 3822: 3693: 3682: 3662: 3650: 3613: 3601: 3585: 3583: 3550: 3538: 3521: 3498: 3487: 3468: 3444: 3435: 3423: 3363:→ ∞, the first example will decay like 1/ 3247: 3241: 3240: 3221: 3215: 3214: 3195: 3194: 3186: 2908: 2902: 2869: 2863: 2842: 2836: 2816: 2793: 2772: 2766: 2747: 2746: 2739: 2738: 2729: 2723: 2690: 2684: 2643: 2637: 2457: 2429: 2417: 2405: 2396: 2370: 2351: 2337: 2303: 2284: 2262: 2251: 2233: 2227: 1919: 1914: 1904: 1888: 1876: 1788: 1746: 1723: 1718: 1706: 1702: 1681: 1658: 1635: 1624: 1603: 1565: 1554: 1533: 1467: 1455: 1428: 1417: 1396: 1314: 1282: 1260: 1253: 1244: 1236: 1217: 1202: 1180: 1170: 1151: 1141: 1138: 1094: 1083: 1051: 1029: 1027: 946: 924: 918: 829: 819: 782: 770: 761: 752: 743: 718: 705: 677: 671: 646: 633: 613: 607: 584: 565: 542: 479: 474: 464: 448: 436: 390: 385: 375: 350: 338: 297: 292: 282: 263: 251: 4724:on , which in waveform terminology is a 4594:(this is the uncertainty in position of 3355:for any even positive Schwartz function 2679:will be complex as well. The expression 2625:I would like to have any mention of the 1505:{\displaystyle E\propto \varphi ''(0)=0} 6235: 5997: 5968: 5632:Relation to independence/ Kac's Theorem 5606: 5563:Yes, I like that book, and cite it in " 5210: 5004: 4655:, so the expected value will be E = (1− 2256: 2222:Characteristic function is an integral 1646:{\displaystyle \varphi (t)=e^{-ct^{4}}} 1576:{\displaystyle \varphi (t)=e^{-ct^{4}}} 1439:{\displaystyle \varphi (t)=e^{-ct^{4}}} 1329: 1308: 147: 49: 19: 3941:≤ 1 for all 1 ≤ p ≤ ∞ (interpolation). 3047:Well, you have to be a little careful 1996:The article gives the impression that 535:Why?! We have (assuming analyticity), 6868:Unknown-priority mathematics articles 6757:. Either way, there is a factor of 2 6574:2601:200:C000:1A0:BC00:5039:DB55:E9EC 6052:This can't be right: F(A) - F(A - 0)? 4789:) is the characteristic function for 3024:Does anybody know if standard normal 1389:The would-be characteristic function 7: 6765:to the inverse Fourier transform? — 6552:of the probability density function. 4602:(the “uncertainty” in “velocity” of 4586:, whereas the c.f. will be equal to 3566:) indeed coincides with its own cf. 2897:without going back to definition of 2112:a real-valued random variable whose 177:This article is within the scope of 92:This article is within the scope of 6878:High-importance Statistics articles 6315:{\displaystyle F_{X}(a-0)=F_{X}(a)} 4582:², then the pdf is proportional to 4180:as a linear map acting on variable 3559:{\displaystyle a\leq (4\pi /3)^{2}} 2050:defined in the real numbers have a 1858:history of characteristic functions 1852:History of Characteristic Functions 38:It is of interest to the following 6156: 5854: 5527:I correct the article accordingly. 5509: 5428: 5098: 4124:The same space, or the dual space? 2252: 1748: 1663: 1212: 14: 6673:{\displaystyle \omega =2\pi \nu } 5623:, see (3.2) in Chapter 2, page 95 5586:I will try to keep that in mind. 4722:uniform distribution (continuous) 4590:. Thus one curve has the “width” 4072:Thank you. You are right. I did. 4051: 2128:, then the answer is yes: either 1778:I suggest you follow the link to 112:Knowledge:WikiProject Mathematics 6805:But most importantly, the CF is 5811:{\displaystyle \mathbb {R} ^{d}} 5694:{\displaystyle \varphi _{(X,Y)}} 5662:to the characteristic functions 5619:Probability: theory and examples 4696:A 'physical' interpretation for 4576:Heisenberg uncertainty principle 4420:characteristic function of a set 4152:then the argument of cf is also 197:Knowledge:WikiProject Statistics 170: 149: 115:Template:WikiProject Mathematics 79: 69: 51: 20: 6883:WikiProject Statistics articles 4920:Why only for strictly positive? 4598:), while the other has width 1/ 3950:Applying our projection to the 1811:{\displaystyle \varphi (t): --> 885:"Different change of parameter" 217:This article has been rated as 200:Template:WikiProject Statistics 6725: 6708: 6610: 6604: 6544:If a random variable admits a 6529:04:25, 24 September 2019 (UTC) 6506: 6494: 6464: 6455: 6418: 6412: 6382: 6368: 6356: 6334:03:48, 24 September 2019 (UTC) 6309: 6303: 6287: 6275: 6232: 6226: 6153: 6139: 6127: 6111: 6105: 5944: 5932: 5926: 5914: 5686: 5674: 5463: 5460: 5454: 5445: 5439: 5433: 5422: 5408: 5402: 5379: 5373: 5350: 5344: 5335: 5329: 5273: 5255: 5236: 5224: 5207: 5201: 5113: 5103: 5095: 5018: 5009: 4979: 4973: 4643:For Bernoulli this is simple: 4349:22:05, 26 September 2010 (UTC) 4335:21:55, 26 September 2010 (UTC) 4285:20:58, 26 September 2010 (UTC) 4247:18:32, 24 September 2009 (UTC) 4138:17:42, 24 September 2009 (UTC) 3954:gives an example to show that 3886: 3836: 3675: 3659: 3634: 3619: 3598: 3588: 3547: 3529: 3480: 3425: 3259: 3253: 3233: 3227: 3207: 3201: 2884: 2875: 2743: 2705: 2696: 2658: 2652: 2514:Matrix-valued random variables 2469: 2463: 2441: 2435: 2418: 2397: 2382: 2376: 2315: 2309: 2245: 2239: 1937: 1931: 1926: 1920: 1894: 1881: 1799: 1793: 1719: 1707: 1692: 1686: 1614: 1608: 1544: 1538: 1493: 1487: 1473: 1460: 1407: 1401: 1326: 1320: 1206: 1192: 1186: 1163: 1157: 1114: 1111: 1105: 1087: 1073: 1067: 1061: 1045: 1039: 967: 952: 936: 930: 847: 841: 836: 830: 816: 806: 800: 794: 789: 783: 695: 689: 684: 678: 581: 568: 553: 547: 497: 491: 486: 480: 454: 441: 408: 402: 397: 391: 372: 362: 356: 343: 315: 309: 304: 298: 269: 256: 1: 6565:sign inside the exponential. 5051:is a probability measure. If 4915:03:45, 19 February 2014 (UTC) 4838:21:10, 18 November 2011 (UTC) 4748:. Perhaps it would help with 3367:, the second will decay like 2968:few drawbacks compared to cf. 2611:13:16, 10 December 2008 (UTC) 2592:09:41, 10 December 2008 (UTC) 2545:08:05, 10 December 2008 (UTC) 2167:16:33, 19 November 2008 (UTC) 2089:15:02, 19 November 2008 (UTC) 2064:12:25, 19 November 2008 (UTC) 2042:10:01, 19 November 2008 (UTC) 2026:19:06, 18 November 2008 (UTC) 1986:22:53, 17 November 2007 (UTC) 1377:10:28, 21 February 2007 (UTC) 191:and see a list of open tasks. 106:and see a list of open tasks. 6863:B-Class mathematics articles 6546:probability density function 5782:{\displaystyle X,Y\in L^{1}} 4879:matrix argument of the c.f. 4871:It was supposed to be small 4052:This can't be right, surely? 3375:. The example suggested by 2528:15:45, 9 December 2008 (UTC) 2134:cardinality of the continuum 1367:13:07, 21 October 2006 (UTC) 1358:22:17, 20 October 2006 (UTC) 1175: 1146: 1034: 879:16:45, 11 October 2009 (UTC) 526:14:00, 11 October 2009 (UTC) 6873:B-Class Statistics articles 4895:19:58, 7 October 2009 (UTC) 4865:08:53, 7 October 2009 (UTC) 4471:22:34, 5 October 2009 (UTC) 4410:09:42, 5 October 2009 (UTC) 4387:00:04, 5 October 2009 (UTC) 4118:15:07, 21 August 2009 (UTC) 4103:13:54, 21 August 2009 (UTC) 3912:that might be interesting. 1669:{\displaystyle \pm \infty } 1009:10:50, 21 August 2009 (UTC) 905:15:56, 20 August 2009 (UTC) 6899: 6512:{\displaystyle F_{X}(a-0)} 5565:Conditioning (probability) 4750:their use in data analysis 4265:lies in the same space as 2627:moment-generating function 2621:Moment generating function 2597:distributions such as the 2565:would be associated with T 2198:20:17, 23 March 2009 (UTC) 2014:standard probability space 1865:19:22, 10 April 2007 (UTC) 1847:08:55, 29 April 2008 (UTC) 1829:08:56, 28 April 2008 (UTC) 1780:positive definite function 1773:08:44, 28 April 2008 (UTC) 1594:09:34, 25 April 2008 (UTC) 1524:09:25, 24 April 2008 (UTC) 6848:19:26, 31 July 2024 (UTC) 6820:15:51, 31 July 2024 (UTC) 6784:19:02, 29 July 2022 (UTC) 6648:. With the substitution 6582:18:51, 29 July 2022 (UTC) 6470:{\displaystyle F_{X}(a-)} 6063:is (possibly) an atom of 5596:21:45, 30 July 2015 (UTC) 5580:21:24, 30 July 2015 (UTC) 5558:21:14, 30 July 2015 (UTC) 5539:21:04, 30 July 2015 (UTC) 5518:{\displaystyle -\infty .} 5356:{\displaystyle F(b)-F(a)} 5305:20:30, 30 July 2015 (UTC) 4953:23:30, 29 July 2015 (UTC) 4938:20:07, 29 July 2015 (UTC) 4227:. Maybe this is what the 3042:18:35, 30 June 2009 (UTC) 3010:08:50, 18 June 2009 (UTC) 2986:21:14, 17 June 2009 (UTC) 2955:13:27, 17 June 2009 (UTC) 2939:11:24, 17 June 2009 (UTC) 2890:{\displaystyle M_{X}(it)} 2718:is not correct. Function 2711:{\displaystyle M_{X}(it)} 2664:{\displaystyle M_{iX}(t)} 2498:21:20, 16 June 2009 (UTC) 1954:Inverse fourier transform 216: 165: 131: 64: 46: 6045:11:01, 30 May 2016 (UTC) 4803:11:56, 26 May 2010 (UTC) 4776:09:17, 26 May 2010 (UTC) 4762:08:11, 26 May 2010 (UTC) 4622:23:58, 25 May 2010 (UTC) 4557:14:26, 25 May 2010 (UTC) 4367:? The reason being that 4082:18:36, 6 July 2009 (UTC) 4067:15:38, 6 July 2009 (UTC) 4032:11:28, 2 July 2009 (UTC) 3785:09:27, 2 July 2009 (UTC) 3733:19:28, 1 July 2009 (UTC) 3390:09:27, 2 July 2009 (UTC) 3312:18:40, 1 July 2009 (UTC) 3282:13:07, 1 July 2009 (UTC) 3161:12:47, 1 July 2009 (UTC) 3099:07:56, 1 July 2009 (UTC) 3065:06:53, 1 July 2009 (UTC) 2132:is countable or has the 1869:Please check your work. 1382:Bochner-Khinchin theorem 134:project's priority scale 5631: 1754:{\displaystyle \infty } 95:WikiProject Mathematics 6751: 6674: 6642: 6513: 6471: 6428: 6316: 6247: 6014: 5885: 5841: 5812: 5783: 5748:(Kac's theorem, from ) 5735: 5695: 5656: 5519: 5493: 5473: 5386: 5357: 5284: 5071: 5070:{\displaystyle a<b} 5045: 5025: 4961:The inversion formula. 3893: 3705: 3560: 3510: 3266: 2918: 2891: 2852: 2825: 2805: 2782: 2755: 2712: 2665: 2482: 2322: 2188:for pathologies here. 1944: 1813: 1755: 1735: 1670: 1647: 1577: 1506: 1440: 1344: 1121: 974: 857: 731: 659: 504: 415: 322: 180:WikiProject Statistics 28:This article is rated 6752: 6675: 6643: 6514: 6472: 6429: 6317: 6248: 6015: 5886: 5842: 5813: 5784: 5736: 5696: 5657: 5567:" and other articles. 5520: 5494: 5474: 5387: 5358: 5285: 5072: 5046: 5026: 4088:Properties - symmetry 3894: 3706: 3561: 3511: 3267: 2919: 2917:{\displaystyle M_{X}} 2892: 2853: 2851:{\displaystyle M_{X}} 2826: 2806: 2783: 2781:{\displaystyle M_{X}} 2756: 2713: 2666: 2483: 2323: 2098:distribution function 2052:distribution function 1998:every random variable 1992:Pathological examples 1945: 1814: 1756: 1736: 1671: 1648: 1578: 1507: 1441: 1345: 1122: 975: 858: 732: 660: 505: 416: 323: 6684: 6652: 6595: 6481: 6477:is better; I guess, 6442: 6343: 6262: 6092: 5895: 5851: 5847:are independent 2. 5825: 5793: 5754: 5705: 5666: 5640: 5503: 5483: 5396: 5385:{\displaystyle F(b)} 5367: 5323: 5083: 5055: 5044:{\displaystyle \mu } 5035: 4967: 4742:WP:original research 4726:rectangular function 4639:) 17 May 2010 (UTC) 3821: 3582: 3520: 3422: 3185: 2901: 2862: 2835: 2815: 2792: 2765: 2722: 2683: 2671:denotes an mgf of a 2636: 2599:Wishart distribution 2336: 2226: 2146:continuum hypothesis 2126:Continuum hypothesis 2096:, yes, but the page 2006:pathological example 1875: 1787: 1745: 1680: 1657: 1602: 1532: 1454: 1395: 1137: 1026: 917: 742: 670: 541: 435: 424:Please compare with 337: 250: 118:mathematics articles 6519:is out of fashion. 6196: 5840:{\displaystyle X,Y} 5655:{\displaystyle X,Y} 5142: 3952:triangular function 3899:is a projection in 3121:Hermite polynomials 3081:(seeing as we need 2120:? If we accept the 2074:Cantor distribution 2010:perforated interval 1930: 1355:Faisel Gulamhussein 1252: 490: 401: 308: 203:Statistics articles 6829:made some edits. — 6825:Quite true! I've 6747: 6670: 6638: 6509: 6467: 6424: 6401: 6312: 6243: 6236: 6176: 6160: 6010: 5998: 5969: 5881: 5837: 5808: 5779: 5731: 5691: 5652: 5515: 5489: 5469: 5432: 5382: 5353: 5280: 5279: 5211: 5125: 5102: 5067: 5041: 5021: 5005: 4706:Fourier transforms 3889: 3834: 3772:theorem of Bochner 3701: 3556: 3506: 3459: 3262: 2914: 2887: 2848: 2821: 2804:{\displaystyle it} 2801: 2778: 2751: 2708: 2661: 2478: 2318: 2257: 1940: 1910: 1808: 1751: 1731: 1666: 1643: 1573: 1502: 1436: 1340: 1330: 1309: 1232: 1216: 1117: 1101: 970: 853: 727: 655: 500: 470: 411: 381: 318: 288: 87:Mathematics portal 34:content assessment 6700: 6550:Fourier transform 6374: 6174: 6145: 6048: 6031:comment added by 5492:{\displaystyle a} 5414: 5250: 5196: 5087: 4855:comment added by 4841: 4824:comment added by 4547:comment added by 4511:(5), you plug in 3969:positive definite 3833: 3768:positive definite 3670: 3644: 3458: 2824:{\displaystyle t} 2079: 1988: 1972:comment added by 1306: 1230: 1198: 1178: 1149: 1079: 1059: 1037: 1015:Inversion theorem 895:comment added by 776: 627: 599: 516:comment added by 237: 236: 233: 232: 229: 228: 144: 143: 140: 139: 6890: 6835: 6771: 6760: 6756: 6754: 6753: 6748: 6740: 6739: 6718: 6701: 6699: 6688: 6679: 6677: 6676: 6671: 6647: 6645: 6644: 6639: 6631: 6630: 6590: 6518: 6516: 6515: 6510: 6493: 6492: 6476: 6474: 6473: 6468: 6454: 6453: 6433: 6431: 6430: 6425: 6411: 6410: 6400: 6355: 6354: 6321: 6319: 6318: 6313: 6302: 6301: 6274: 6273: 6252: 6250: 6249: 6244: 6225: 6224: 6215: 6214: 6195: 6187: 6175: 6173: 6162: 6159: 6126: 6125: 6104: 6103: 6047: 6025: 6019: 6017: 6016: 6011: 6009: 6008: 5988: 5980: 5979: 5959: 5951: 5950: 5902: 5890: 5888: 5887: 5882: 5880: 5879: 5874: 5846: 5844: 5843: 5838: 5817: 5815: 5814: 5809: 5807: 5806: 5801: 5788: 5786: 5785: 5780: 5778: 5777: 5740: 5738: 5737: 5732: 5730: 5729: 5717: 5716: 5700: 5698: 5697: 5692: 5690: 5689: 5661: 5659: 5658: 5653: 5624: 5622: 5615:Durrett, Richard 5611: 5524: 5522: 5521: 5516: 5498: 5496: 5495: 5490: 5478: 5476: 5475: 5470: 5431: 5391: 5389: 5388: 5383: 5362: 5360: 5359: 5354: 5289: 5287: 5286: 5281: 5251: 5243: 5197: 5195: 5187: 5186: 5185: 5164: 5163: 5144: 5141: 5136: 5124: 5123: 5101: 5076: 5074: 5073: 5068: 5050: 5048: 5047: 5042: 5030: 5028: 5027: 5022: 5003: 5002: 4893: 4889: 4886: 4867: 4840: 4818: 4693: 4689: 4686: 4677:complex analysis 4620: 4616: 4613: 4559: 4533: 4529: 4526: 4503:(0) is, plug in 4491:, so it has the 4469: 4465: 4462: 4385: 4381: 4378: 4283: 4279: 4276: 4245: 4241: 4238: 3898: 3896: 3895: 3890: 3885: 3884: 3879: 3878: 3868: 3867: 3862: 3861: 3851: 3850: 3835: 3826: 3710: 3708: 3707: 3702: 3700: 3699: 3698: 3697: 3671: 3663: 3658: 3657: 3645: 3643: 3618: 3617: 3607: 3606: 3605: 3586: 3572:I thought about 3565: 3563: 3562: 3557: 3555: 3554: 3542: 3515: 3513: 3512: 3507: 3505: 3504: 3503: 3502: 3473: 3472: 3460: 3457: 3446: 3440: 3439: 3271: 3269: 3268: 3263: 3252: 3251: 3246: 3245: 3226: 3225: 3220: 3219: 3200: 3199: 2923: 2921: 2920: 2915: 2913: 2912: 2896: 2894: 2893: 2888: 2874: 2873: 2857: 2855: 2854: 2849: 2847: 2846: 2830: 2828: 2827: 2822: 2810: 2808: 2807: 2802: 2787: 2785: 2784: 2779: 2777: 2776: 2760: 2758: 2757: 2752: 2750: 2742: 2734: 2733: 2717: 2715: 2714: 2709: 2695: 2694: 2670: 2668: 2667: 2662: 2651: 2650: 2601:. Best wishes, 2487: 2485: 2484: 2479: 2462: 2461: 2434: 2433: 2421: 2416: 2415: 2400: 2389: 2385: 2375: 2374: 2362: 2361: 2327: 2325: 2324: 2319: 2308: 2307: 2295: 2294: 2273: 2272: 2255: 2238: 2237: 2077: 1967: 1949: 1947: 1946: 1941: 1929: 1918: 1909: 1908: 1893: 1892: 1819: 1816: 1815: 1809: 1760: 1758: 1757: 1752: 1740: 1738: 1737: 1732: 1730: 1729: 1728: 1727: 1722: 1710: 1675: 1673: 1672: 1667: 1652: 1650: 1649: 1644: 1642: 1641: 1640: 1639: 1582: 1580: 1579: 1574: 1572: 1571: 1570: 1569: 1511: 1509: 1508: 1503: 1486: 1472: 1471: 1445: 1443: 1442: 1437: 1435: 1434: 1433: 1432: 1349: 1347: 1346: 1341: 1319: 1318: 1307: 1305: 1297: 1296: 1295: 1274: 1273: 1254: 1251: 1243: 1231: 1229: 1218: 1215: 1185: 1184: 1179: 1171: 1156: 1155: 1150: 1142: 1126: 1124: 1123: 1118: 1100: 1099: 1098: 1060: 1052: 1038: 1030: 1007: 1004: 996: 993: 986:Fourier analysis 979: 977: 976: 971: 951: 950: 929: 928: 907: 862: 860: 859: 854: 840: 839: 824: 823: 793: 792: 777: 775: 774: 762: 757: 756: 736: 734: 733: 728: 723: 722: 710: 709: 688: 687: 664: 662: 661: 656: 651: 650: 638: 637: 628: 626: 618: 617: 608: 600: 598: 590: 589: 588: 566: 528: 509: 507: 506: 501: 489: 478: 469: 468: 453: 452: 420: 418: 417: 412: 400: 389: 380: 379: 355: 354: 331:I changed it to 327: 325: 324: 319: 307: 296: 287: 286: 268: 267: 223:importance scale 205: 204: 201: 198: 195: 174: 167: 166: 161: 153: 146: 120: 119: 116: 113: 110: 89: 84: 83: 73: 66: 65: 55: 48: 31: 25: 24: 16: 6898: 6897: 6893: 6892: 6891: 6889: 6888: 6887: 6853: 6852: 6831: 6792: 6767: 6763:closely related 6758: 6728: 6692: 6682: 6681: 6650: 6649: 6613: 6593: 6592: 6588: 6537: 6521:Boris Tsirelson 6484: 6479: 6478: 6445: 6440: 6439: 6436:one-sided limit 6402: 6346: 6341: 6340: 6293: 6265: 6260: 6259: 6216: 6197: 6166: 6117: 6095: 6090: 6089: 6072: 6054: 6026: 6020: 5989: 5960: 5903: 5893: 5892: 5869: 5849: 5848: 5823: 5822: 5796: 5791: 5790: 5769: 5752: 5751: 5721: 5708: 5703: 5702: 5669: 5664: 5663: 5638: 5637: 5634: 5629: 5628: 5627: 5613: 5612: 5608: 5572:Boris Tsirelson 5531:Boris Tsirelson 5501: 5500: 5481: 5480: 5394: 5393: 5365: 5364: 5321: 5320: 5297:Boris Tsirelson 5188: 5168: 5146: 5145: 5112: 5081: 5080: 5053: 5052: 5033: 5032: 4988: 4965: 4964: 4930:Boris Tsirelson 4922: 4903: 4887: 4884: 4880: 4850: 4847: 4819: 4687: 4684: 4680: 4614: 4611: 4607: 4542: 4527: 4524: 4520: 4463: 4460: 4456: 4438: 4402:Boris Tsirelson 4379: 4376: 4372: 4357: 4341:Boris Tsirelson 4327:Boris Tsirelson 4319: 4312: 4306: 4298: 4277: 4274: 4270: 4239: 4236: 4232: 4219:arbitrary, and 4130:Boris Tsirelson 4126: 4110:Boris Tsirelson 4090: 4074:Boris Tsirelson 4054: 3940: 3872: 3855: 3819: 3818: 3689: 3678: 3646: 3609: 3608: 3597: 3587: 3580: 3579: 3546: 3518: 3517: 3494: 3483: 3464: 3450: 3431: 3420: 3419: 3304:Boris Tsirelson 3239: 3213: 3183: 3182: 3153:Boris Tsirelson 3086: 3022: 3020:Standard normal 2904: 2899: 2898: 2865: 2860: 2859: 2838: 2833: 2832: 2813: 2812: 2790: 2789: 2768: 2763: 2762: 2725: 2720: 2719: 2686: 2681: 2680: 2639: 2634: 2633: 2623: 2572: 2568: 2564: 2516: 2453: 2425: 2401: 2366: 2347: 2343: 2339: 2334: 2333: 2299: 2280: 2258: 2229: 2224: 2223: 2190:Boris Tsirelson 2122:Axiom of Choice 2102:so pathological 2048:random variable 1994: 1956: 1900: 1884: 1873: 1872: 1854: 1784: 1783: 1743: 1742: 1717: 1698: 1678: 1677: 1655: 1654: 1631: 1620: 1600: 1599: 1598:On any compact 1561: 1550: 1530: 1529: 1479: 1463: 1452: 1451: 1424: 1413: 1393: 1392: 1384: 1310: 1298: 1278: 1256: 1255: 1222: 1169: 1140: 1135: 1134: 1090: 1024: 1023: 1017: 999: 994: 991: 942: 920: 915: 914: 890: 887: 871:Boris Tsirelson 825: 815: 778: 766: 748: 740: 739: 714: 701: 673: 668: 667: 642: 629: 619: 609: 591: 580: 567: 539: 538: 511: 460: 444: 433: 432: 371: 346: 335: 334: 278: 259: 248: 247: 242: 219:High-importance 202: 199: 196: 193: 192: 160:High‑importance 159: 117: 114: 111: 108: 107: 85: 78: 32:on Knowledge's 29: 12: 11: 5: 6896: 6894: 6886: 6885: 6880: 6875: 6870: 6865: 6855: 6854: 6851: 6850: 6791: 6788: 6787: 6786: 6746: 6743: 6738: 6735: 6731: 6727: 6724: 6721: 6717: 6713: 6710: 6707: 6704: 6698: 6695: 6691: 6669: 6666: 6663: 6660: 6657: 6637: 6634: 6629: 6626: 6623: 6620: 6616: 6612: 6609: 6606: 6603: 6600: 6536: 6533: 6532: 6531: 6508: 6505: 6502: 6499: 6496: 6491: 6487: 6466: 6463: 6460: 6457: 6452: 6448: 6423: 6420: 6417: 6414: 6409: 6405: 6399: 6396: 6393: 6390: 6387: 6384: 6381: 6377: 6373: 6370: 6367: 6364: 6361: 6358: 6353: 6349: 6311: 6308: 6305: 6300: 6296: 6292: 6289: 6286: 6283: 6280: 6277: 6272: 6268: 6256: 6255: 6254: 6253: 6242: 6239: 6234: 6231: 6228: 6223: 6219: 6213: 6210: 6207: 6204: 6200: 6194: 6191: 6186: 6183: 6179: 6172: 6169: 6165: 6158: 6155: 6152: 6148: 6144: 6141: 6138: 6135: 6132: 6129: 6124: 6120: 6116: 6113: 6110: 6107: 6102: 6098: 6084: 6083: 6070: 6053: 6050: 6007: 6004: 6001: 5996: 5992: 5987: 5983: 5978: 5975: 5972: 5967: 5963: 5958: 5954: 5949: 5946: 5943: 5940: 5937: 5934: 5931: 5928: 5925: 5922: 5919: 5916: 5913: 5910: 5906: 5901: 5878: 5873: 5868: 5865: 5862: 5859: 5856: 5836: 5833: 5830: 5820: 5805: 5800: 5776: 5772: 5768: 5765: 5762: 5759: 5728: 5724: 5720: 5715: 5711: 5688: 5685: 5682: 5679: 5676: 5672: 5651: 5648: 5645: 5633: 5630: 5626: 5625: 5605: 5604: 5600: 5599: 5598: 5584: 5583: 5582: 5568: 5544: 5543: 5542: 5541: 5528: 5525: 5514: 5511: 5508: 5488: 5468: 5465: 5462: 5459: 5456: 5453: 5450: 5447: 5444: 5441: 5438: 5435: 5430: 5427: 5424: 5421: 5417: 5413: 5410: 5407: 5404: 5401: 5381: 5378: 5375: 5372: 5352: 5349: 5346: 5343: 5340: 5337: 5334: 5331: 5328: 5310: 5309: 5308: 5307: 5294: 5293: 5292: 5291: 5290: 5278: 5275: 5272: 5269: 5266: 5263: 5260: 5257: 5254: 5249: 5246: 5241: 5238: 5235: 5232: 5229: 5226: 5223: 5220: 5217: 5214: 5209: 5206: 5203: 5200: 5194: 5191: 5184: 5181: 5178: 5175: 5171: 5167: 5162: 5159: 5156: 5153: 5149: 5140: 5135: 5132: 5128: 5122: 5119: 5115: 5111: 5108: 5105: 5100: 5097: 5094: 5090: 5066: 5063: 5060: 5040: 5020: 5017: 5014: 5011: 5008: 5001: 4998: 4995: 4991: 4987: 4984: 4981: 4978: 4975: 4972: 4921: 4918: 4902: 4899: 4898: 4897: 4846: 4843: 4810: 4809: 4808: 4807: 4806: 4805: 4778: 4629: 4628: 4627: 4626: 4625: 4624: 4563: 4562: 4561: 4560: 4549:87.119.246.143 4535: 4534: 4476: 4475: 4474: 4473: 4434: 4413: 4412: 4356: 4353: 4352: 4351: 4337: 4317: 4310: 4307:? The dual to 4304: 4296: 4291: 4250: 4249: 4197: 4125: 4122: 4121: 4120: 4089: 4086: 4085: 4084: 4059:90.206.183.244 4053: 4050: 4049: 4048: 4047: 4046: 4045: 4044: 4043: 4042: 4041: 4040: 4039: 4038: 4037: 4036: 4035: 4034: 4001: 4000: 3999: 3998: 3997: 3996: 3995: 3994: 3993: 3992: 3991: 3990: 3989: 3988: 3987: 3986: 3985: 3984: 3978: 3972: 3962: 3948: 3942: 3938: 3931: 3925: 3919: 3888: 3883: 3877: 3871: 3866: 3860: 3854: 3849: 3844: 3841: 3838: 3832: 3829: 3800: 3799: 3798: 3797: 3796: 3795: 3794: 3793: 3792: 3791: 3790: 3789: 3788: 3787: 3746: 3745: 3744: 3743: 3742: 3741: 3740: 3739: 3738: 3737: 3736: 3735: 3713: 3712: 3711: 3696: 3692: 3688: 3685: 3681: 3677: 3674: 3669: 3666: 3661: 3656: 3653: 3649: 3642: 3639: 3636: 3633: 3630: 3627: 3624: 3621: 3616: 3612: 3604: 3600: 3596: 3593: 3590: 3570: 3553: 3549: 3545: 3541: 3537: 3534: 3531: 3528: 3525: 3501: 3497: 3493: 3490: 3486: 3482: 3479: 3476: 3471: 3467: 3463: 3456: 3453: 3449: 3443: 3438: 3434: 3430: 3427: 3405: 3404: 3403: 3402: 3401: 3400: 3399: 3398: 3397: 3396: 3395: 3394: 3393: 3392: 3323: 3322: 3321: 3320: 3319: 3318: 3317: 3316: 3315: 3314: 3291: 3290: 3289: 3288: 3287: 3286: 3285: 3284: 3261: 3258: 3255: 3250: 3244: 3238: 3235: 3232: 3229: 3224: 3218: 3212: 3209: 3206: 3203: 3198: 3193: 3190: 3168: 3167: 3166: 3165: 3164: 3163: 3104: 3103: 3102: 3101: 3084: 3068: 3067: 3021: 3018: 3017: 3016: 3015: 3014: 3013: 3012: 2969: 2965: 2958: 2957: 2911: 2907: 2886: 2883: 2880: 2877: 2872: 2868: 2845: 2841: 2820: 2800: 2797: 2775: 2771: 2749: 2745: 2741: 2737: 2732: 2728: 2707: 2704: 2701: 2698: 2693: 2689: 2673:complex-valued 2660: 2657: 2654: 2649: 2646: 2642: 2622: 2619: 2618: 2617: 2616: 2615: 2614: 2613: 2581: 2580: 2579: 2578: 2577: 2574: 2570: 2566: 2562: 2550: 2549: 2548: 2547: 2515: 2512: 2511: 2510: 2509: 2508: 2507: 2506: 2505: 2504: 2503: 2502: 2501: 2500: 2477: 2474: 2471: 2468: 2465: 2460: 2456: 2452: 2449: 2446: 2443: 2440: 2437: 2432: 2428: 2424: 2420: 2414: 2411: 2408: 2404: 2399: 2395: 2392: 2388: 2384: 2381: 2378: 2373: 2369: 2365: 2360: 2357: 2354: 2350: 2346: 2342: 2317: 2314: 2311: 2306: 2302: 2298: 2293: 2290: 2287: 2283: 2279: 2276: 2271: 2268: 2265: 2261: 2254: 2250: 2247: 2244: 2241: 2236: 2232: 2209: 2208: 2207: 2206: 2205: 2204: 2203: 2202: 2201: 2200: 2176: 2175: 2174: 2173: 2172: 2171: 2170: 2169: 1993: 1990: 1955: 1952: 1939: 1936: 1933: 1928: 1925: 1922: 1917: 1913: 1907: 1903: 1899: 1896: 1891: 1887: 1883: 1880: 1853: 1850: 1832: 1831: 1807: 1804: 1801: 1798: 1795: 1792: 1750: 1726: 1721: 1716: 1713: 1709: 1705: 1701: 1697: 1694: 1691: 1688: 1685: 1665: 1662: 1638: 1634: 1630: 1627: 1623: 1619: 1616: 1613: 1610: 1607: 1568: 1564: 1560: 1557: 1553: 1549: 1546: 1543: 1540: 1537: 1501: 1498: 1495: 1492: 1489: 1485: 1482: 1478: 1475: 1470: 1466: 1462: 1459: 1431: 1427: 1423: 1420: 1416: 1412: 1409: 1406: 1403: 1400: 1383: 1380: 1351: 1350: 1339: 1336: 1333: 1328: 1325: 1322: 1317: 1313: 1304: 1301: 1294: 1291: 1288: 1285: 1281: 1277: 1272: 1269: 1266: 1263: 1259: 1250: 1247: 1242: 1239: 1235: 1228: 1225: 1221: 1214: 1211: 1208: 1205: 1201: 1197: 1194: 1191: 1188: 1183: 1177: 1174: 1168: 1165: 1162: 1159: 1154: 1148: 1145: 1128: 1127: 1116: 1113: 1110: 1107: 1104: 1097: 1093: 1089: 1086: 1082: 1078: 1075: 1072: 1069: 1066: 1063: 1058: 1055: 1050: 1047: 1044: 1041: 1036: 1033: 1016: 1013: 1012: 1011: 982: 981: 980: 969: 966: 963: 960: 957: 954: 949: 945: 941: 938: 935: 932: 927: 923: 897:130.207.104.54 886: 883: 882: 881: 867: 866: 865: 864: 863: 852: 849: 846: 843: 838: 835: 832: 828: 822: 818: 814: 811: 808: 805: 802: 799: 796: 791: 788: 785: 781: 773: 769: 765: 760: 755: 751: 747: 737: 726: 721: 717: 713: 708: 704: 700: 697: 694: 691: 686: 683: 680: 676: 665: 654: 649: 645: 641: 636: 632: 625: 622: 616: 612: 606: 603: 597: 594: 587: 583: 579: 576: 573: 570: 564: 561: 558: 555: 552: 549: 546: 530: 529: 499: 496: 493: 488: 485: 482: 477: 473: 467: 463: 459: 456: 451: 447: 443: 440: 410: 407: 404: 399: 396: 393: 388: 384: 378: 374: 370: 367: 364: 361: 358: 353: 349: 345: 342: 317: 314: 311: 306: 303: 300: 295: 291: 285: 281: 277: 274: 271: 266: 262: 258: 255: 241: 238: 235: 234: 231: 230: 227: 226: 215: 209: 208: 206: 189:the discussion 175: 163: 162: 154: 142: 141: 138: 137: 130: 124: 123: 121: 104:the discussion 91: 90: 74: 62: 61: 56: 44: 43: 37: 26: 13: 10: 9: 6: 4: 3: 2: 6895: 6884: 6881: 6879: 6876: 6874: 6871: 6869: 6866: 6864: 6861: 6860: 6858: 6849: 6845: 6841: 6837: 6834: 6828: 6824: 6823: 6822: 6821: 6817: 6813: 6808: 6803: 6799: 6795: 6789: 6785: 6781: 6777: 6773: 6770: 6764: 6744: 6741: 6736: 6733: 6729: 6722: 6719: 6715: 6711: 6705: 6702: 6696: 6693: 6689: 6680:that becomes 6667: 6664: 6661: 6658: 6655: 6635: 6632: 6627: 6624: 6621: 6618: 6614: 6607: 6601: 6598: 6586: 6585: 6584: 6583: 6579: 6575: 6571: 6566: 6564: 6560: 6555: 6553: 6551: 6547: 6540: 6534: 6530: 6526: 6522: 6503: 6500: 6497: 6489: 6485: 6461: 6458: 6450: 6446: 6437: 6421: 6415: 6407: 6403: 6397: 6394: 6391: 6388: 6385: 6379: 6371: 6365: 6362: 6359: 6351: 6347: 6338: 6337: 6336: 6335: 6331: 6327: 6323: 6306: 6298: 6294: 6290: 6284: 6281: 6278: 6270: 6266: 6240: 6237: 6229: 6221: 6217: 6211: 6208: 6205: 6202: 6198: 6192: 6189: 6184: 6181: 6177: 6170: 6167: 6163: 6150: 6142: 6136: 6133: 6130: 6122: 6118: 6114: 6108: 6100: 6096: 6088: 6087: 6086: 6085: 6081: 6077: 6076: 6075: 6073: 6066: 6062: 6058: 6049: 6046: 6042: 6038: 6034: 6030: 6024: 6005: 6002: 5999: 5994: 5990: 5981: 5976: 5973: 5970: 5965: 5961: 5952: 5941: 5938: 5935: 5929: 5923: 5920: 5917: 5908: 5904: 5876: 5866: 5863: 5860: 5857: 5834: 5831: 5828: 5819: 5803: 5774: 5770: 5766: 5763: 5760: 5757: 5749: 5746: 5742: 5726: 5722: 5718: 5713: 5709: 5683: 5680: 5677: 5670: 5649: 5646: 5643: 5620: 5616: 5610: 5607: 5603: 5597: 5593: 5589: 5585: 5581: 5577: 5573: 5569: 5566: 5562: 5561: 5559: 5555: 5551: 5546: 5545: 5540: 5536: 5532: 5529: 5526: 5512: 5506: 5486: 5466: 5457: 5451: 5448: 5442: 5436: 5425: 5419: 5411: 5405: 5399: 5376: 5370: 5347: 5341: 5338: 5332: 5326: 5318: 5314: 5313: 5312: 5311: 5306: 5302: 5298: 5295: 5276: 5267: 5264: 5261: 5252: 5247: 5244: 5239: 5233: 5230: 5227: 5221: 5218: 5215: 5212: 5204: 5198: 5192: 5189: 5182: 5179: 5176: 5173: 5169: 5165: 5160: 5157: 5154: 5151: 5147: 5138: 5133: 5130: 5126: 5120: 5117: 5109: 5106: 5092: 5079: 5078: 5064: 5061: 5058: 5038: 5015: 5012: 5006: 4999: 4996: 4993: 4989: 4985: 4982: 4976: 4970: 4962: 4959: 4958: 4956: 4955: 4954: 4950: 4946: 4942: 4941: 4940: 4939: 4935: 4931: 4927: 4926:User:Manoguru 4919: 4917: 4916: 4912: 4908: 4907:OldMacDonalds 4900: 4896: 4891: 4890: 4878: 4874: 4870: 4869: 4868: 4866: 4862: 4858: 4854: 4844: 4842: 4839: 4835: 4831: 4827: 4823: 4814: 4804: 4800: 4796: 4792: 4788: 4783: 4779: 4777: 4773: 4769: 4765: 4764: 4763: 4759: 4755: 4751: 4747: 4743: 4739: 4735: 4731: 4727: 4723: 4719: 4715: 4711: 4707: 4703: 4699: 4695: 4694: 4691: 4690: 4678: 4674: 4670: 4666: 4662: 4658: 4654: 4650: 4646: 4642: 4641: 4640: 4638: 4634: 4623: 4618: 4617: 4605: 4601: 4597: 4593: 4589: 4585: 4581: 4577: 4573: 4569: 4568: 4567: 4566: 4565: 4564: 4558: 4554: 4550: 4546: 4539: 4538: 4537: 4536: 4531: 4530: 4518: 4514: 4510: 4506: 4502: 4498: 4494: 4490: 4486: 4482: 4481: 4480: 4472: 4467: 4466: 4454: 4450: 4446: 4442: 4437: 4433: 4429: 4425: 4421: 4417: 4416: 4415: 4414: 4411: 4407: 4403: 4399: 4395: 4391: 4390: 4389: 4388: 4383: 4382: 4370: 4366: 4362: 4354: 4350: 4346: 4342: 4338: 4336: 4332: 4328: 4324: 4320: 4313: 4303: 4299: 4292: 4289: 4288: 4287: 4286: 4281: 4280: 4268: 4264: 4260: 4255: 4248: 4243: 4242: 4230: 4226: 4222: 4218: 4214: 4210: 4206: 4202: 4198: 4195: 4191: 4187: 4186:inner product 4183: 4179: 4175: 4171: 4167: 4163: 4159: 4155: 4151: 4147: 4142: 4141: 4140: 4139: 4135: 4131: 4123: 4119: 4115: 4111: 4107: 4106: 4105: 4104: 4100: 4096: 4087: 4083: 4079: 4075: 4071: 4070: 4069: 4068: 4064: 4060: 4033: 4029: 4025: 4021: 4017: 4016: 4015: 4014: 4013: 4012: 4011: 4010: 4009: 4008: 4007: 4006: 4005: 4004: 4003: 4002: 3982: 3979: 3976: 3973: 3970: 3966: 3963: 3961: 3957: 3953: 3949: 3946: 3943: 3936: 3932: 3929: 3926: 3923: 3920: 3917: 3914: 3913: 3911: 3907: 3902: 3881: 3869: 3864: 3852: 3842: 3839: 3830: 3827: 3816: 3815: 3814: 3813: 3812: 3811: 3810: 3809: 3808: 3807: 3806: 3805: 3804: 3803: 3802: 3801: 3786: 3782: 3778: 3773: 3769: 3765: 3760: 3759: 3758: 3757: 3756: 3755: 3754: 3753: 3752: 3751: 3750: 3749: 3748: 3747: 3734: 3730: 3726: 3722: 3718: 3714: 3694: 3690: 3686: 3683: 3679: 3672: 3667: 3664: 3654: 3651: 3647: 3640: 3637: 3631: 3628: 3625: 3622: 3614: 3610: 3602: 3594: 3591: 3578: 3577: 3575: 3571: 3569: 3551: 3543: 3539: 3535: 3532: 3526: 3523: 3499: 3495: 3491: 3488: 3484: 3477: 3474: 3469: 3465: 3461: 3454: 3451: 3447: 3441: 3436: 3432: 3428: 3417: 3416: 3415: 3414: 3413: 3412: 3411: 3410: 3409: 3408: 3407: 3406: 3391: 3387: 3383: 3378: 3374: 3370: 3366: 3362: 3358: 3354: 3350: 3346: 3342: 3337: 3336: 3335: 3334: 3333: 3332: 3331: 3330: 3329: 3328: 3327: 3326: 3325: 3324: 3313: 3309: 3305: 3301: 3300: 3299: 3298: 3297: 3296: 3295: 3294: 3293: 3292: 3283: 3279: 3275: 3256: 3248: 3236: 3230: 3222: 3210: 3204: 3191: 3188: 3180: 3176: 3175: 3174: 3173: 3172: 3171: 3170: 3169: 3162: 3158: 3154: 3150: 3146: 3142: 3138: 3134: 3130: 3126: 3122: 3118: 3114: 3110: 3109: 3108: 3107: 3106: 3105: 3100: 3096: 3092: 3088: 3080: 3076: 3072: 3071: 3070: 3069: 3066: 3062: 3058: 3054: 3050: 3046: 3045: 3044: 3043: 3039: 3035: 3031: 3027: 3019: 3011: 3007: 3003: 2999: 2994: 2989: 2988: 2987: 2983: 2979: 2974: 2970: 2966: 2962: 2961: 2960: 2959: 2956: 2952: 2948: 2943: 2942: 2941: 2940: 2936: 2932: 2928: 2925: 2909: 2905: 2881: 2878: 2870: 2866: 2843: 2839: 2818: 2798: 2795: 2773: 2769: 2735: 2730: 2726: 2702: 2699: 2691: 2687: 2678: 2674: 2655: 2647: 2644: 2640: 2630: 2628: 2620: 2612: 2608: 2604: 2600: 2595: 2594: 2593: 2589: 2585: 2582: 2575: 2569:rather than T 2560: 2559: 2558: 2557: 2554: 2553: 2552: 2551: 2546: 2542: 2538: 2534: 2533: 2532: 2531: 2530: 2529: 2525: 2521: 2513: 2499: 2495: 2491: 2475: 2472: 2466: 2458: 2454: 2450: 2447: 2444: 2438: 2430: 2426: 2422: 2412: 2409: 2406: 2402: 2393: 2390: 2386: 2379: 2371: 2367: 2363: 2358: 2355: 2352: 2348: 2344: 2340: 2331: 2312: 2304: 2300: 2296: 2291: 2288: 2285: 2281: 2277: 2274: 2269: 2266: 2263: 2259: 2248: 2242: 2234: 2230: 2221: 2220: 2219: 2218: 2217: 2216: 2215: 2214: 2213: 2212: 2211: 2210: 2199: 2195: 2191: 2186: 2185: 2184: 2183: 2182: 2181: 2180: 2179: 2178: 2177: 2168: 2164: 2160: 2159:Darth Albmont 2155: 2151: 2147: 2143: 2139: 2135: 2131: 2127: 2123: 2119: 2115: 2111: 2107: 2103: 2099: 2095: 2092: 2091: 2090: 2086: 2082: 2075: 2071: 2067: 2066: 2065: 2061: 2057: 2056:Darth Albmont 2053: 2049: 2045: 2044: 2043: 2039: 2035: 2030: 2029: 2028: 2027: 2023: 2019: 2015: 2011: 2007: 2003: 1999: 1991: 1989: 1987: 1983: 1979: 1975: 1971: 1963: 1960: 1953: 1951: 1934: 1923: 1915: 1911: 1905: 1901: 1897: 1889: 1885: 1878: 1870: 1867: 1866: 1863: 1862:Michael Stone 1859: 1851: 1849: 1848: 1844: 1840: 1835: 1830: 1826: 1822: 1805: 1802: 1796: 1790: 1781: 1777: 1776: 1775: 1774: 1770: 1766: 1761: 1724: 1714: 1711: 1703: 1699: 1695: 1689: 1683: 1660: 1636: 1632: 1628: 1625: 1621: 1617: 1611: 1605: 1596: 1595: 1591: 1587: 1566: 1562: 1558: 1555: 1551: 1547: 1541: 1535: 1526: 1525: 1521: 1517: 1512: 1499: 1496: 1490: 1483: 1480: 1476: 1468: 1464: 1457: 1449: 1446: 1429: 1425: 1421: 1418: 1414: 1410: 1404: 1398: 1390: 1387: 1381: 1379: 1378: 1375: 1374:BenWilliamson 1369: 1368: 1365: 1360: 1359: 1356: 1337: 1334: 1331: 1323: 1315: 1311: 1302: 1299: 1292: 1289: 1286: 1283: 1279: 1275: 1270: 1267: 1264: 1261: 1257: 1248: 1245: 1240: 1237: 1233: 1226: 1223: 1219: 1209: 1203: 1195: 1189: 1181: 1172: 1166: 1160: 1152: 1143: 1133: 1132: 1131: 1108: 1102: 1095: 1091: 1084: 1076: 1070: 1064: 1056: 1053: 1048: 1042: 1031: 1022: 1021: 1020: 1014: 1010: 1005: 998: 997: 987: 983: 964: 961: 958: 955: 947: 943: 939: 933: 925: 921: 913: 912: 910: 909: 908: 906: 902: 898: 894: 884: 880: 876: 872: 868: 850: 844: 833: 826: 820: 812: 809: 803: 797: 786: 779: 771: 767: 763: 758: 753: 749: 745: 738: 724: 719: 715: 711: 706: 702: 698: 692: 681: 674: 666: 652: 647: 643: 639: 634: 630: 623: 620: 614: 610: 604: 601: 595: 592: 585: 577: 574: 571: 562: 559: 556: 550: 544: 537: 536: 534: 533: 532: 531: 527: 523: 519: 518:188.36.27.144 515: 494: 483: 475: 471: 465: 461: 457: 449: 445: 438: 430: 429: 428: 426: 422: 405: 394: 386: 382: 376: 368: 365: 359: 351: 347: 340: 332: 329: 312: 301: 293: 289: 283: 279: 275: 272: 264: 260: 253: 245: 239: 224: 220: 214: 211: 210: 207: 190: 186: 182: 181: 176: 173: 169: 168: 164: 158: 155: 152: 148: 135: 129: 126: 125: 122: 105: 101: 97: 96: 88: 82: 77: 75: 72: 68: 67: 63: 60: 57: 54: 50: 45: 41: 35: 27: 23: 18: 17: 6832: 6806: 6804: 6800: 6796: 6793: 6768: 6762: 6569: 6567: 6562: 6558: 6556: 6543: 6541: 6538: 6324: 6257: 6079: 6068: 6064: 6060: 6056: 6055: 6027:— Preceding 6021: 5747: 5744: 5743: 5635: 5621:(Second ed.) 5618: 5609: 5601: 5316: 4960: 4923: 4904: 4882: 4876: 4872: 4857:161.53.64.70 4848: 4820:— Preceding 4815: 4811: 4790: 4786: 4745: 4737: 4733: 4713: 4712:= ½ and 1 – 4709: 4701: 4697: 4682: 4672: 4668: 4664: 4660: 4656: 4652: 4648: 4644: 4630: 4609: 4603: 4599: 4595: 4591: 4587: 4583: 4579: 4571: 4522: 4516: 4512: 4508: 4504: 4500: 4496: 4484: 4477: 4458: 4452: 4448: 4444: 4440: 4435: 4431: 4427: 4423: 4397: 4393: 4374: 4368: 4364: 4360: 4358: 4315: 4308: 4301: 4294: 4272: 4266: 4262: 4258: 4253: 4251: 4234: 4224: 4220: 4216: 4212: 4208: 4204: 4200: 4193: 4189: 4181: 4177: 4173: 4169: 4165: 4161: 4157: 4153: 4149: 4145: 4127: 4091: 4055: 4019: 3980: 3974: 3964: 3959: 3955: 3944: 3934: 3927: 3921: 3915: 3909: 3905: 3900: 3720: 3716: 3567: 3372: 3368: 3364: 3360: 3356: 3352: 3348: 3344: 3340: 3178: 3148: 3144: 3140: 3136: 3132: 3128: 3124: 3116: 3112: 3082: 3078: 3074: 3052: 3048: 3025: 3023: 2997: 2992: 2972: 2929: 2926: 2676: 2672: 2631: 2624: 2517: 2329: 2153: 2149: 2141: 2137: 2129: 2117: 2114:sample space 2109: 2105: 2101: 2093: 2069: 1997: 1995: 1964: 1961: 1957: 1871: 1868: 1855: 1836: 1833: 1762: 1597: 1527: 1513: 1450: 1447: 1391: 1388: 1385: 1370: 1361: 1352: 1129: 1018: 989: 888: 423: 333: 330: 246: 243: 218: 178: 93: 40:WikiProjects 6812:165.89.30.1 6559:all of them 4851:—Preceding 4826:Zenmaster82 4633:User:NoName 4543:—Preceding 4495:. That is, 4363:instead of 4300:, not just 4184:, or as an 3721:physicist's 3030:fixed point 2811:instead of 2046:Does every 2002:almost sure 1968:—Preceding 1818:0.}" /: --> 891:—Preceding 512:—Preceding 240:Discussion… 109:Mathematics 100:mathematics 59:Mathematics 6857:Categories 6082:is scalar: 5602:References 4592:&sigma 4229:dual space 4215:² we have 4168:then also 3079:N(0, (2π)) 194:Statistics 185:statistics 157:Statistics 6326:Manybytes 4160:. And if 4024:Thenub314 3777:Thenub314 3382:Thenub314 3274:Thenub314 3181:and take 3057:Thenub314 2964:analysis. 2081:Baccyak4H 1786:0.}": --> 1364:Thenub314 6844:contribs 6836:uantling 6780:contribs 6772:uantling 6563:negative 6438:. Well, 6041:contribs 6029:unsigned 5617:(1996), 5588:Manoguru 5550:Manoguru 5363:but not 4945:Manoguru 4853:unsigned 4834:contribs 4822:unsigned 4782:referent 4768:Melcombe 4637:contribs 4545:unsigned 4517:argument 4493:argument 4489:function 4355:Notation 4323:Lp space 4314:is some 3967:must be 3002:Melcombe 2947:Melcombe 2584:Melcombe 2520:Melcombe 2034:Melcombe 2016:. Darth 2012:in page 1982:contribs 1974:Nwerneck 1970:unsigned 1839:MagnusPI 1821:Melcombe 1765:MagnusPI 1586:Melcombe 1516:MagnusPI 893:unsigned 514:unsigned 6570:inverse 6074:) then 6057:Theorem 5745:Theorem 4252:So, if 4095:Rlendog 3725:Stpasha 3715:(where 3516:(where 3343:; sech( 3143:, –1. – 3135:, –1. – 3127:, –1. – 3115:, –1. – 3091:Stpasha 3034:Stpasha 2978:Stpasha 2931:Stpasha 2490:Stpasha 2018:Albmont 869:Right? 221:on the 30:B-class 6827:boldly 5750:. Let 5031:where 4875:— the 4667:= 1 − 4483:Well, 4321:, see 4207:² and 3049:N(0,1) 3026:N(0,1) 2603:Robinh 2561:each X 2537:Robinh 2110:define 36:scale. 6797:: --> 6059:. If 6033:Ceacy 5077:then 4888:pasha 4688:pasha 4615:pasha 4528:pasha 4464:pasha 4430:) or 4380:pasha 4278:pasha 4240:pasha 3764:Boris 3719:is a 3574:Boris 3377:Boris 3139:, 1, 3131:, 1, 3087:(0)=1 1803:: --> 1130:then 995:pasha 6840:talk 6816:talk 6776:talk 6578:talk 6525:talk 6434:See 6395:< 6330:talk 6258:But 6037:talk 5701:and 5592:talk 5576:talk 5554:talk 5535:talk 5301:talk 5062:< 4963:Let 4949:talk 4934:talk 4911:talk 4861:talk 4830:talk 4799:talk 4795:Qwfp 4772:talk 4758:talk 4754:Qwfp 4738:kind 4730:sinc 4651:and 4553:talk 4447:and 4406:talk 4392:And 4345:talk 4331:talk 4231:is. 4134:talk 4114:talk 4099:talk 4078:talk 4063:talk 4028:talk 3781:talk 3729:talk 3386:talk 3308:talk 3278:talk 3157:talk 3095:talk 3061:talk 3038:talk 3006:talk 2982:talk 2951:talk 2935:talk 2607:talk 2588:talk 2541:talk 2524:talk 2494:talk 2194:talk 2163:talk 2085:Yak! 2070:e.g. 2060:talk 2038:talk 2022:talk 1978:talk 1843:talk 1825:talk 1769:talk 1590:talk 1520:talk 1003:talk 901:talk 875:talk 522:talk 213:High 6807:not 6376:lim 6147:lim 6078:If 5891:, 5821:1. 5789:be 5499:to 5416:lim 5317:all 5089:lim 4928:.) 4877:k×p 4746:use 4740:of 4718:cos 4681:// 4679:. 4608:// 4521:// 4271:// 4178:t′X 3904:in 3347:); 2116:is 1812:0.} 1200:lim 1081:lim 510:. 128:??? 6859:: 6846:) 6842:| 6818:) 6782:) 6778:| 6745:ω 6734:ω 6723:π 6712:ω 6703:∫ 6697:π 6668:ν 6665:π 6656:ω 6636:ν 6625:ν 6622:π 6608:ν 6599:∫ 6580:) 6554:" 6527:) 6501:− 6462:− 6383:→ 6363:− 6332:) 6282:− 6218:φ 6203:− 6182:− 6178:∫ 6157:∞ 6154:→ 6134:− 6115:− 6043:) 6039:• 6006:η 6003:⋅ 5982:⋅ 5977:ξ 5974:⋅ 5948:⟩ 5942:η 5936:ξ 5912:⟨ 5867:∈ 5864:ξ 5858:η 5855:∀ 5767:∈ 5723:φ 5710:φ 5671:φ 5594:) 5578:) 5556:) 5537:) 5510:∞ 5507:− 5449:− 5429:∞ 5426:− 5423:→ 5339:− 5303:) 5253:μ 5222:μ 5199:φ 5174:− 5166:− 5152:− 5131:− 5127:∫ 5118:− 5110:π 5099:∞ 5096:→ 5039:μ 5007:μ 4986:∫ 4971:φ 4951:) 4936:) 4913:) 4892:» 4885:st 4881:… 4863:) 4836:) 4832:• 4801:) 4774:) 4760:) 4692:» 4685:st 4673:pe 4671:+ 4665:pe 4663:+ 4619:» 4612:st 4555:) 4532:» 4525:st 4468:» 4461:st 4457:… 4422:: 4408:) 4384:» 4377:st 4373:… 4347:) 4333:) 4325:. 4282:» 4275:st 4269:. 4244:» 4237:st 4233:… 4192:, 4136:) 4116:) 4101:) 4080:) 4065:) 4030:) 3937:|| 3933:|| 3783:) 3731:) 3687:π 3684:− 3668:π 3629:− 3592:− 3536:π 3527:≤ 3492:π 3489:− 3455:π 3442:− 3388:) 3310:) 3280:) 3159:) 3097:) 3063:) 3040:) 3008:) 2984:) 2953:) 2937:) 2744:→ 2609:) 2590:) 2571:ji 2567:ij 2563:ij 2543:) 2526:) 2496:) 2448:∫ 2394:∫ 2391:≤ 2345:∫ 2332:: 2278:∫ 2231:ϕ 2196:) 2165:) 2087:) 2072:, 2062:) 2054:? 2040:) 2024:) 1984:) 1980:• 1950:. 1912:φ 1845:) 1837:-- 1827:) 1806:0. 1791:φ 1771:) 1763:-- 1749:∞ 1725:α 1704:− 1684:φ 1664:∞ 1661:± 1626:− 1606:φ 1592:) 1556:− 1536:φ 1522:) 1514:-- 1481:φ 1477:∝ 1419:− 1399:φ 1353:-- 1312:φ 1284:− 1276:− 1262:− 1249:τ 1241:τ 1238:− 1234:∫ 1227:π 1213:∞ 1207:→ 1204:τ 1176:¯ 1167:− 1147:¯ 1096:− 1088:→ 1035:¯ 1000:» 992:st 962:π 956:− 944:φ 922:φ 903:) 877:) 827:ϕ 810:− 780:ϕ 675:ϕ 605:∑ 563:∑ 545:ϕ 524:) 472:φ 427:. 421:. 383:φ 366:− 328:. 290:φ 276:− 6838:( 6833:Q 6814:( 6774:( 6769:Q 6759:π 6742:d 6737:t 6730:e 6726:) 6720:2 6716:/ 6709:( 6706:F 6694:2 6690:1 6662:2 6659:= 6633:d 6628:t 6619:2 6615:e 6611:) 6605:( 6602:F 6589:π 6576:( 6542:" 6523:( 6507:) 6504:0 6498:a 6495:( 6490:X 6486:F 6465:) 6459:a 6456:( 6451:X 6447:F 6422:. 6419:) 6416:b 6413:( 6408:X 6404:F 6398:a 6392:b 6389:, 6386:a 6380:b 6372:= 6369:) 6366:0 6360:a 6357:( 6352:X 6348:F 6328:( 6310:) 6307:a 6304:( 6299:X 6295:F 6291:= 6288:) 6285:0 6279:a 6276:( 6271:X 6267:F 6241:t 6238:d 6233:) 6230:t 6227:( 6222:X 6212:a 6209:t 6206:i 6199:e 6193:T 6190:+ 6185:T 6171:T 6168:2 6164:1 6151:T 6143:= 6140:) 6137:0 6131:a 6128:( 6123:X 6119:F 6112:) 6109:a 6106:( 6101:X 6097:F 6080:X 6071:X 6069:F 6065:X 6061:a 6035:( 6000:Y 5995:i 5991:e 5986:E 5971:X 5966:i 5962:e 5957:E 5953:= 5945:) 5939:, 5933:( 5930:, 5927:) 5924:Y 5921:, 5918:X 5915:( 5909:i 5905:e 5900:E 5877:d 5872:R 5861:, 5835:Y 5832:, 5829:X 5804:d 5799:R 5775:1 5771:L 5764:Y 5761:, 5758:X 5727:Y 5719:, 5714:X 5687:) 5684:Y 5681:, 5678:X 5675:( 5650:Y 5647:, 5644:X 5590:( 5574:( 5552:( 5533:( 5513:. 5487:a 5467:. 5464:) 5461:) 5458:a 5455:( 5452:F 5446:) 5443:b 5440:( 5437:F 5434:( 5420:a 5412:= 5409:) 5406:b 5403:( 5400:F 5380:) 5377:b 5374:( 5371:F 5351:) 5348:a 5345:( 5342:F 5336:) 5333:b 5330:( 5327:F 5299:( 5277:. 5274:) 5271:} 5268:b 5265:, 5262:a 5259:{ 5256:( 5248:2 5245:1 5240:+ 5237:) 5234:b 5231:, 5228:a 5225:( 5219:= 5216:t 5213:d 5208:) 5205:t 5202:( 5193:t 5190:i 5183:b 5180:t 5177:i 5170:e 5161:a 5158:t 5155:i 5148:e 5139:T 5134:T 5121:1 5114:) 5107:2 5104:( 5093:T 5065:b 5059:a 5019:) 5016:x 5013:d 5010:( 5000:x 4997:t 4994:i 4990:e 4983:= 4980:) 4977:t 4974:( 4947:( 4932:( 4909:( 4873:t 4859:( 4828:( 4797:( 4791:U 4787:t 4770:( 4756:( 4734:t 4732:( 4714:p 4710:p 4702:t 4698:t 4669:p 4661:e 4659:) 4657:p 4653:p 4649:p 4645:X 4635:( 4604:x 4600:σ 4596:x 4588:e 4584:e 4580:σ 4572:t 4551:( 4513:t 4509:φ 4505:t 4501:φ 4497:t 4485:R 4453:ϕ 4449:φ 4445:ϕ 4441:χ 4436:A 4432:1 4428:A 4426:( 4424:1 4404:( 4398:χ 4394:χ 4369:φ 4365:φ 4361:χ 4343:( 4329:( 4318:q 4316:L 4311:p 4309:L 4305:2 4302:L 4297:p 4295:L 4267:X 4263:t 4259:H 4254:t 4225:X 4221:t 4217:X 4213:L 4211:∈ 4209:t 4205:L 4203:∈ 4201:X 4194:X 4190:t 4188:( 4182:X 4174:R 4172:∈ 4170:t 4166:R 4164:∈ 4162:X 4158:R 4156:∈ 4154:t 4150:R 4148:∈ 4146:X 4132:( 4112:( 4097:( 4076:( 4061:( 4026:( 4020:e 3981:ƒ 3975:ƒ 3965:ƒ 3960:x 3956:ƒ 3945:ƒ 3939:p 3935:ƒ 3928:ƒ 3922:ƒ 3916:ƒ 3910:ƒ 3906:L 3901:L 3887:) 3882:3 3876:F 3870:+ 3865:2 3859:F 3853:+ 3848:F 3843:+ 3840:I 3837:( 3831:4 3828:1 3779:( 3727:( 3717:H 3695:2 3691:x 3680:e 3676:) 3673:x 3665:2 3660:( 3655:n 3652:2 3648:H 3641:! 3638:! 3635:) 3632:1 3626:n 3623:2 3620:( 3615:n 3611:2 3603:n 3599:) 3595:1 3589:( 3552:2 3548:) 3544:3 3540:/ 3533:4 3530:( 3524:a 3500:2 3496:x 3485:e 3481:) 3478:1 3475:+ 3470:2 3466:x 3462:a 3452:2 3448:3 3437:4 3433:x 3429:a 3426:( 3384:( 3373:ƒ 3369:e 3365:x 3361:x 3357:ƒ 3353:ƒ 3351:* 3349:ƒ 3345:x 3341:e 3306:( 3276:( 3260:) 3257:f 3254:( 3249:3 3243:F 3237:+ 3234:) 3231:f 3228:( 3223:2 3217:F 3211:+ 3208:) 3205:f 3202:( 3197:F 3192:+ 3189:f 3179:ƒ 3155:( 3149:x 3145:i 3141:i 3137:i 3133:i 3129:i 3125:i 3117:i 3113:i 3093:( 3085:X 3083:f 3075:φ 3059:( 3053:π 3036:( 3004:( 2998:t 2993:t 2980:( 2973:t 2949:( 2933:( 2910:X 2906:M 2885:) 2882:t 2879:i 2876:( 2871:X 2867:M 2844:X 2840:M 2819:t 2799:t 2796:i 2774:X 2770:M 2748:R 2740:R 2736:: 2731:X 2727:M 2706:) 2703:t 2700:i 2697:( 2692:X 2688:M 2677:t 2659:) 2656:t 2653:( 2648:X 2645:i 2641:M 2605:( 2586:( 2573:, 2539:( 2522:( 2492:( 2476:1 2473:= 2470:) 2467:x 2464:( 2459:X 2455:F 2451:d 2445:= 2442:) 2439:x 2436:( 2431:X 2427:F 2423:d 2419:| 2413:x 2410:t 2407:i 2403:e 2398:| 2387:| 2383:) 2380:x 2377:( 2372:X 2368:F 2364:d 2359:x 2356:t 2353:i 2349:e 2341:| 2330:t 2316:) 2313:x 2310:( 2305:X 2301:F 2297:d 2292:x 2289:t 2286:i 2282:e 2275:= 2270:X 2267:t 2264:i 2260:e 2253:E 2249:= 2246:) 2243:t 2240:( 2235:X 2192:( 2161:( 2154:c 2150:2 2142:X 2138:X 2130:X 2118:X 2106:X 2083:( 2058:( 2036:( 2020:( 1976:( 1938:) 1935:0 1932:( 1927:) 1924:n 1921:( 1916:X 1906:n 1902:i 1898:= 1895:) 1890:n 1886:X 1882:( 1879:E 1841:( 1823:( 1800:) 1797:t 1794:( 1767:( 1720:| 1715:t 1712:c 1708:| 1700:e 1696:= 1693:) 1690:t 1687:( 1637:4 1633:t 1629:c 1622:e 1618:= 1615:) 1612:t 1609:( 1588:( 1567:4 1563:t 1559:c 1552:e 1548:= 1545:) 1542:t 1539:( 1518:( 1500:0 1497:= 1494:) 1491:0 1488:( 1484:″ 1474:] 1469:2 1465:X 1461:[ 1458:E 1430:4 1426:t 1422:c 1415:e 1411:= 1408:) 1405:t 1402:( 1338:. 1335:t 1332:d 1327:) 1324:t 1321:( 1316:X 1303:t 1300:i 1293:y 1290:t 1287:i 1280:e 1271:x 1268:t 1265:i 1258:e 1246:+ 1224:2 1220:1 1210:+ 1196:= 1193:) 1190:x 1187:( 1182:X 1173:F 1164:) 1161:y 1158:( 1153:X 1144:F 1115:] 1112:) 1109:y 1106:( 1103:F 1092:x 1085:y 1077:+ 1074:) 1071:x 1068:( 1065:F 1062:[ 1057:2 1054:1 1049:= 1046:) 1043:x 1040:( 1032:F 1006:» 968:) 965:t 959:2 953:( 948:1 940:= 937:) 934:t 931:( 926:2 899:( 873:( 851:. 848:) 845:0 842:( 837:) 834:n 831:( 821:n 817:) 813:i 807:( 804:= 801:) 798:0 795:( 790:) 787:n 784:( 772:n 768:i 764:1 759:= 754:n 750:X 746:E 725:; 720:n 716:X 712:E 707:n 703:i 699:= 696:) 693:0 690:( 685:) 682:n 679:( 653:; 648:n 644:X 640:E 635:n 631:i 624:! 621:n 615:n 611:t 602:= 596:! 593:n 586:n 582:) 578:X 575:t 572:i 569:( 560:E 557:= 554:) 551:t 548:( 520:( 498:) 495:0 492:( 487:) 484:n 481:( 476:X 466:n 462:i 458:= 455:) 450:n 446:X 442:( 439:E 409:) 406:0 403:( 398:) 395:n 392:( 387:X 377:n 373:) 369:i 363:( 360:= 357:) 352:n 348:X 344:( 341:E 316:) 313:0 310:( 305:) 302:n 299:( 294:X 284:n 280:i 273:= 270:) 265:n 261:X 257:( 254:E 225:. 136:. 42::

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