Sorry, does the article actually give reasons why the bell curve is "everywhere"?

For simplicity, take N identically distributed random variables that are uniform on the interval from [-1/2,1/2], so the probability distribution function, f(x), on the interval from [-1/2,1/2] is 1.

The Fourier transform of f(x), F(w), is essentially sin(w)/w. Taking only the first few terms of the Taylor expansion, ignoring constants, gives (1-w^2).

Convolution is multiplication in Fourier space, so you get (1-w^2)^n. Squinting, (1-w^2)^n ~ (1-n w^2 / n)^n ~ exp(-n w^2). The Fourier transform of a Gaussian is a Gaussian, so the result holds.

Unfortunately I haven't worked it out myself but I've been told if you fiddle with the exponent of 2 (presumably choosing it to be in the range of (0,2]), this gives the motivation for Levy stable distributions, which is another way to see why fat-tailed/Levy stable distributions are so ubiquitous.

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woopsn an hour ago | parent | next [-]

There's a paragraph on discovery that multinomial distributions are normal in the limit. The turn from there to CLT is not great, but that's a standard way to introduce normal distributions and explains a myriad of statistics.

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WCSTombs 4 hours ago | parent | prev | next [-]

It's not super hard to prove the central limit theorem, and you gave the flavor of one such proof, but it's still a bit much for the likely audience of this article, who can't be assumed to have the math background needed to appreciate the argument. And I think you're on the right track with the comment about stable distributions.

	▲	abetusk 3 hours ago \| parent [-]
		The Fourier transform of a uniform distribution is the sinc function which looks like a quadratic locally around 0. Convolution to multiplication is how the quadratic goes from downstairs to upstairs, giving the Gaussian. Widths of different uniform distributions along with different centers all still have a quadratic center, so the above argument only needs to be minimally changed. The added bonus is that if the (1-w^2)^n is replaced by (1-w^a)^n, you can sort of see how to get at the Levy stable distribution (see the characteristic function definition [0]). The point is that this gives a simple, high-level motivation as to why it's so common. Aside from seeing this flavor of proof in "An Invitation to Modern Number Theory" [1], I haven't really seen it elsewhere (though, to be fair, I'm not a mathematician). I also have never heard the connection of this method to the Levy stable distributions but for someone communicating it to me personally. I disagree about the audience for Quanta. They tend to be exposed to higher level concepts even if they don't have a lot of in depth experience with them. [0] https://en.wikipedia.org/wiki/Stable_distribution#Parametriz... [1] https://www.amazon.com/Invitation-Modern-Number-Theory/dp/06...

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4 hours ago | parent | prev [-]

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