The Central Limit Theorem postulates a set of N independent random variates will have their probability distribution (P) close to that of a Normal Distribution if N is large.

Its derivation can be found in:

Central Limit Theorem -- from Wolfram MathWorld

and includes an Inverse Fourier Transform of the probability distribution P. But, as described in (Fourier Transform -- from Wolfram MathWorld), the inverse Fourier Transform is a generalization of the Fourier series.

My question is:How can we apply the inverse Fourier Transform to a Probability Distribtuion if this is not a Fourier Series?