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

(2)

is normally distributed with  and .

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 .

			(3)
			(4)
			(5)
			(6)

Now write

(7)

so we have

			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)

Now expand

(17)

so

			(18)
			(19)
			(20)

since

			(21)
			(22)

Taking the Fourier transform,

			(23)
			(24)

This is of the form

(25)

where  and . But this is a Fourier transform of a Gaussian function, so

(26)

(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.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

Feller, W. "The Fundamental Limit Theorems in Probability." Bull. Amer. Math. Soc. 51, 800-832, 1945.

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, p. 229, 1968.

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971.

Kallenberg, O. Foundations of Modern Probability. New York: Springer-Verlag, 1997.

Lindeberg, J. W. "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung." Math. Z. 15, 211-225, 1922.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 112-113, 1992.

Trotter, H. F. "An Elementary Proof of the Central Limit Theorem." Arch. Math. 10, 226-234, 1959.

Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483-494, 1995.