Feller-Lévy Condition

Given a sequence of independent random variates $X_1$, $X_2$, ..., if ${\sigma_k}^2=\mathop{\rm var}\nolimits (X_k)$ and

$${\rho_n}^2\equiv\max_{k\leq n}\left({{\sigma_k}^2\over{s_n}^2}\right),$$

then

$$\lim_{n\to\infty} {\rho_n}^2=0.$$

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.

References

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