Lindeberg Condition

A Sufficient condition on the Lindeberg-Feller Central Limit Theorem. Given random variates $X_1$, $X_2$, ..., let $\left\langle{X_i}\right\rangle{}=0$, the Variance ${\sigma_i}^2$ of $X_i$ be finite, and Variance of the distribution consisting of a sum of $X_i$s

$$S_n\equiv X_1+X_2+\ldots+X_n$$ (1)

be

$${s_n}^2\equiv\sum_{i=1}^n {\sigma_i}^2.$$ (2)

Let

$$\Lambda_n(\epsilon)\equiv \sum_{k=1}^n \left\langle{\left.{\... ...ht\vert {\vert X_k\vert\over s_n}\geq\epsilon}\right\rangle{},$$ (3)

then the Lindeberg condition is

$$\lim_{n\to\infty} \Lambda_n(\epsilon) =0$$ (4)

for all $\epsilon>0$.

See also Feller-Lévy Condition

References

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