Lindeberg-Feller Central Limit Theorem

If the random variates $X_1$, $X_2$, ... satisfy the Lindeberg Condition, then for all $a<b$,

$$\lim_{n\to\infty} P\left({a<{S_n\over s_n}<b}\right)=\Phi(b)-\Phi(a),$$

where $\Phi$ is the Normal Distribution Function.

See also Central Limit Theorem, Feller-Lévy Condition, Normal Distribution Function

References

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