Let
be a set of independent random variates and each have an arbitrary probability
distribution
with mean and a finite variance . Then the normal form variate

(1)

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 and variance . If conversion to normal form is not performed, then
the variate

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 .

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

(27)

(28)

(29)

But
and ,
so

(30)

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