Feller-Lévy Condition

Given a sequence of independent random variates , , ..., if  and

then

This means that if the Lindeberg condition holds for the sequence of variates , ..., then the variance of an individual term in the sum  of  is asymptotically negligible. For such sequences, the Lindeberg condition is necessary as well as sufficient for the Lindeberg-Feller central limit theorem to hold.

REFERENCES:

Lindeberg, J. W. "Eine neue Herleitung des Exponentialgesetzes in der Wahrschienlichkeitsrechnung." Math. Z. 15, 211-225, 1922.

Zabell, S. L. "Alan Turing and the Central Limit Theorem." Amer. Math. Monthly 102, 483-494, 1995.