Lindeberg Condition

A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates , , ..., let , the variance  of  be finite, and variance of the distribution consisting of a sum of s

(1)

be

(2)

In the terminology of Zabell (1995), let

(3)

where  denotes the expectation value of  restricted to outcomes , then the Lindeberg condition is

(4)

for all  (Zabell 1995).

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

(5)

or equivalently

(6)

Then the distribution

(7)

tends to the normal distribution with zero expectation and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the individual variances  are small compared to their sum  in the sense that for given  for all sufficiently large ,  for , ...,  (Feller 1971, p. 256).

REFERENCES:

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.