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



F_f^(-1)[P_X(f)](x) approx exp{N[(2piif)/N<x>-((2pif)^2)/(2N^2)<x^2>+1/2((2piif)^2)/(N^2)<x>^2+O(N^(-3))]}
 approx exp[2piifmu_x-((2pif)^2sigma_x^2)/(2N)],



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

Berry-Esséen Theorem, Fourier Transform--Gaussian, Lindeberg Condition, Lindeberg-Feller Central Limit Theorem, Lyapunov Condition Explore this topic in the MathWorld classroom

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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.

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Central Limit Theorem

Cite this as:

Weisstein, Eric W. "Central Limit Theorem." From MathWorld--A Wolfram Web Resource.

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