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Lindeberg-Feller theorem

From Encyclopedia of Mathematics
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A theorem that establishes necessary and sufficient conditions for the asymptotic normality of the distribution function of sums of independent random variables that have finite variances. Let be a sequence of independent random variables with means and finite variances not all of which are zero. Let

In order that

and

for any as , it is necessary and sufficient that the following condition (the Lindeberg condition) is satisfied:

as for any . Sufficiency was proved by J.W. Lindeberg [1] and necessity by W. Feller [2].

References

[1] J.W. Lindeberg, "Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung" Math. Z. , 15 (1922) pp. 211–225
[2] W. Feller, "Ueber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung" Math. Z. , 40 (1935) pp. 521–559
[3] M. Loève, "Probability theory" , Springer (1977)
[4] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
How to Cite This Entry:
Lindeberg-Feller theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lindeberg-Feller_theorem&oldid=22747
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article