### Iterated Convolutions

Any reasonable'' probability density function (PDF) (§C.1.3) has a Fourier transform that looks like near its tip. Iterating convolutions then corresponds to , which becomes [2]

 (D.27)

for large , by the definition of [264]. This proves that the th power of approaches the Gaussian function defined in §D.1 for large .

Since the inverse Fourier transform of a Gaussian is another Gaussian (§D.8), we can define a time-domain function as being sufficiently regular'' when its Fourier transform approaches in a sufficiently small neighborhood of . That is, the Fourier transform simply needs a sufficiently smooth peak'' at that can be expanded into a convergent Taylor series. This obviously holds for the DTFT of any discrete-time window function (the subject of Chapter 3), because the window transform is a finite sum of continuous cosines of the form in the zero-phase case, and complex exponentials in the causal case, each of which is differentiable any number of times in .

Next Section:
Binomial Distribution
Previous Section:
Central Limit Theorem