Lindeberg-Feller theorem

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 $ X _ {1} , X _ {2} \dots $ be a sequence of independent random variables with means $ a _ {1} , a _ {2} \dots $ and finite variances $ \sigma _ {1} ^ {2} , \sigma _ {2} ^ {2} \dots $ not all of which are zero. Let

$$ B _ {n} ^ {2} = \sum_{j=1}^ { n } \sigma _ {j} ^ {2} ,\ \ V _ {j} ( x) = {\mathsf P} \{ x _ {j} < x \} . $$

In order that

$$ B _ {n} ^ {-2} \max _ {1 \leq j \leq n } \ \sigma _ {j} ^ {2} \rightarrow 0 $$

and

$$ {\mathsf P} \left \{ B _ {n} ^ {-1} \sum_{j=1}^ { n } ( X _ {i} - a _ {j} ) < x \right \} \rightarrow \ \frac{1}{\sqrt {2 \pi }} \int\limits _ {- \infty } ^ { x } e ^ {- t ^ {2} /2 } d t $$

for any $ x $ as $ n \rightarrow \infty $, it is necessary and sufficient that the following condition (the Lindeberg condition) is satisfied:

$$ B _ {n} ^ {-2} \sum_{j=1}^ { n } \int\limits _ {| x - a _ {j} | \geq \epsilon B _ {n} } ( x - a _ {j} ) ^ {2} d V _ {j} ( x) \rightarrow 0 $$

as $ n \rightarrow \infty $ for any $ \epsilon > 0 $. 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)