The Central Limit Theorem (Derivation)

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?**

Re: The Central Limit Theorem (Derivation)

Hey Schiavo.

I think you are referring to a common probability transformation known as the characteristic function:

Characteristic function (probability theory) - Wikipedia, the free encyclopedia

Check out the properties and desired constraints for probability functions to get a better answer to your question.

Re: The Central Limit Theorem (Derivation)

Re: The Central Limit Theorem (Derivation)

I saw your link about characteristic functions, but I'm still puzzled. What, exactly is a characteristic function? Is it an approximation of the probability density function? Why would someone prefere working with characteristic functions rather then with probability density function? And, still, if it involves a Fourier transform, how can one "Fourier transform", something that's not a Fourier series?

Re: The Central Limit Theorem (Derivation)

Its basically a transform in probability.

Transforms in probability are used to get systematic ways of finding moments, proving that distributions have a specific form (like adding two Normals is a Normal, adding two independenc Poissons is a Poisson, etc) and also to find probability mass functions and probability density functions given more general situations (like adding say discrete uniform with a binomial).

Things like finding distributions of complex random variable combinations is something where transforms come into play a lot. For example Y = f(X) = X^3 + Z^2 + 5W is an example where transforms play a role (assume X, W, Z independent).

Being able to find moments helps get the critical central moments (mean, variance, kurtosis, skewness) and if this is for an estimator, then this is useful for statistical inference.