 What are characteristic functions? They're like a Fourier transform of the probability density function. If the pdf of a random variable X is f, then its char. func. is φ_{X}(t) = ∫e^{itx}f(x) dx
 They have the exponentialish property (uncoincidentally reminiscent of how the Fourier transform turns convolution into pointwise multiplication) that if X and Y are independent random variables, then φ_{X+Y}(t) = φ_{X}(t)φ_{Y}(t)
 The first few terms of the Taylor series of the characteristic function of a random variable can be worked out just from its moments about zero: consider the defining formula, differentiate n times wrt t, and set t to 0. But for a factor of n! this is the nth derivative, and staring you in the face is a moment integral i^{n}∫x^{n}f(x) dx
 In particular, if your mean is zero, and your standard deviation is one, your char. func.'s Taylor series starts off 1  t^{2}/2 + ...
 If you have n of these dudes all independent, then adding them all up yields a nway product of their characteristic functions, which are all identical, so it's just an nth power.
 After some normalization magic, this winds up resembling the formula (1  t^{2}/n)^{n}
 So, dig deep and remember the limiting formula for the exponential, and realize the char. func. of the result is e^{t2}. This so happens to be that of the standard normal, QED

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Something that's bugged me for a long time is this: How many paths, starting at the origin, taking N steps either up, down, left or right, end up at…

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Still sad that SAC seems to end up being as complicated as it is. Surely there's some deeper duality between…

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I had already been meaning to dig into JaneSt's "Incremental" library, which bills itself as a practical implementation (in ocaml) of the ideas in…
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