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).
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.
Monthly102, 483-494, 1995.
, 
, ..., let 
, the variance 
of 
be finite, and variance of
the distribution consisting of a sum of 
s
denotes the expectation value of 
restricted to outcomes 
, then the Lindeberg condition is
(Zabell 1995).
,
are small compared to their sum 
in the sense that for given 
for all sufficiently
large 
,
for 
,
..., 
(Feller 1971, p. 256).
