# College Math Teaching

## September 2, 2014

### Using convolutions and Fourier Transforms to prove the Central Limit Theorem

Filed under: probability — Tags: , , — collegemathteaching @ 5:40 pm

I’ve used the presentation in the our Probability and Statistics text; it is appropriate given that many of our students haven’t seen the Fourier Transform. But this presentation is excellent.

Upshot: use the convolution to derive the density function for $S_n = X_1 + X_2 + ....X_n$ (independent, identically distributed random variables of finite variance), assume mean is zero, variance is 1 and divide $S_n$ by $\sqrt{n}$ to obtain the variance of the sum to be 1. Then use the Fourier transform on the whole thing (the normalized version) to turn convolution into products, use the definition of Fourier transform and use the Taylor series for the $e^{i 2 \pi x \frac{s}{\sqrt{n}}}$ terms, discard the high order terms, take the limit as $n$ goes to infinity and obtain a Gaussian, which, of course, inverse Fourier transforms to another Gaussian.