Lyapunov Condition

If the third Moment exists for a Distribution of $x_i$ and the Lebesgue Integral is given by

$${r_n}^3 =\sum_{i=1}^n \int_{-\infty}^\infty \vert x\vert^3\,dF_i(x),$$

then if

$$\lim_{n\to\infty} {r_n\over s_n}=0,$$

the Central Limit Theorem holds.

See also Central Limit Theorem