- 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 n-way product of their characteristic functions, which are all identical, so it's just an n-th 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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