Central Limit Theorem

Let X_1,X_2,...,X_N be a set of N independent random variates and each X_i have an arbitrary probability distribution P(x_1,...,x_N) with mean mu_i and a finite variance sigma_i^2. Then the normal form variate

has a limiting cumulative distribution function which approaches a normal distribution.

Under additional conditions on the distribution of the addend, the probability density itself is also normal (Feller 1971) with mean mu=0 and variance sigma^2=1. If conversion to normal form is not performed, then the variate

is normally distributed with mu_X=mu_x and sigma_X=sigma_x/sqrt(N).

Kallenberg (1997) gives a six-line proof of the central limit theorem. For an elementary, but slightly more cumbersome proof of the central limit theorem, consider the inverse Fourier transform of P_X(f).

Now write

so we have

Now expand

so

since

Taking the Fourier transform,

This is of the form

where a=2pi(mu_x-x) and b=(2pisigma_x)^2/2N. But this is a Fourier transform of a Gaussian function, so

(e.g., Abramowitz and Stegun 1972, p. 302, equation 7.4.6). Therefore,

But sigma_X=sigma_x/sqrt(N) and mu_X=mu_x, so

The "fuzzy" central limit theorem says that data which are influenced by many small and unrelated random effects are approximately normally distributed.

See also

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References

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Cite this as:

Weisstein, Eric W. "Central Limit Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CentralLimitTheorem.html

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