Jason (jcreed) wrote,
Jason
jcreed

Reddit's front page had a link to this image complaining about the liberal application of "and trivially we get" in WP's proof of the central limit theorem by characteristic functions. While I do agree that the article could probably be improved (and I don't think I could personally, else I would) there is a really, really cute proof lurking behind those words that I get the gist of, at least. Here is a non-encyclopedic-style summary:

  1. 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 φX(t) = ∫eitxf(x) dx
  2. 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 φX+Y(t) = φX(t)φY(t)
  3. 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 in∫xnf(x) dx
  4. In particular, if your mean is zero, and your standard deviation is one, your char. func.'s Taylor series starts off 1 - t2/2 + ...
  5. 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.
  6. After some normalization magic, this winds up resembling the formula (1 - t2/n)n
  7. 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
Tags: math
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