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Feller-Lévy Condition


Given a sequence of independent random variates X_1, X_2, ..., if sigma_k^2=var(X_k) and

 rho_n^2=max_(k<=n)((sigma_k^2)/(s_n^2)),

then

 lim_(n->infty)rho_n^2=0.

This means that if the Lindeberg condition holds for the sequence of variates X_1, ..., then the variance of an individual term in the sum S_n of X_k is asymptotically negligible. For such sequences, the Lindeberg condition is necessary as well as sufficient for the Lindeberg-Feller central limit theorem to hold.


See also

Berry-Esséen Theorem, Central Limit Theorem, Lindeberg Condition

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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.

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Feller-Lévy Condition

Cite this as:

Weisstein, Eric W. "Feller-Lévy Condition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Feller-LevyCondition.html

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