Lindeberg Condition

A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates X_1, X_2, ..., let <X_i>=0, the variance sigma_i^2 of X_i be finite, and variance of the distribution consisting of a sum of X_is




In the terminology of Zabell (1995), let


where <f:g> denotes the expectation value of f restricted to outcomes g, then the Lindeberg condition is


for all epsilon>0 (Zabell 1995).

In the terminology of Feller (1971), the Lindeberg condition assumed that for each t>0,


or equivalently


Then the distribution


tends to the normal distribution with zero expectation and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the individual variances sigma_k^2 are small compared to their sum s_n^2 in the sense that for given epsilon>0 for all sufficiently large n, sigma_k/s_n<epsilon for k=1, ..., n (Feller 1971, p. 256).

See also

Central Limit Theorem, Feller-Lévy Condition

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Feller, W. "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung." Math. Zeit. 40, 521-559, 1935.Feller, W. "Über den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung, II." Math. Zeit. 42, 301-312, 1935.Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, pp. 257-258, 1971.Lindeberg, J. W. "Eine neue Herleitung des Exponential-gesetzes in der Wahrscheinlichkeitsrechnung." Math. Zeit. 15, 211-235, 1922.Trotter, H. F. "An Elementary Proof of the Central Limit Theorem." Arch. Math. 10, 226-234, 1959.Wallace, D. L. "Asymptotic Approximations to Distributions." Ann. Math. Stat. 29, 635-654, 1958.Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483-494, 1995.

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Lindeberg Condition

Cite this as:

Weisstein, Eric W. "Lindeberg Condition." From MathWorld--A Wolfram Web Resource.

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