# Lindeberg-Feller theorem

From Encyclopedia of Mathematics

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