General Central Limit Theorem says:
Let $ \{(X_{n,j} , 1 ≤ j ≤ n), n ≥ 1\} $ be a triangular array of rowwise independent random variables, set $ S_n = \sum_{j=1}^n X_{n,j}, s_n^2 = \sum_{j=1}^n σ_{n,j}^2,n ≥ 1 $, where $σ_{n,j}^2 = \mathrm{Var}X_{n,j} , 1 ≤ j ≤ n$, and suppose, without restriction, that
$$ \mathrm{E}X_{n,j} = 0 \mbox{ for } \; 1 ≤ j ≤ n, n ≥ 1 $$
and that
$$ s_n^2 = 1 \;\mbox{ for all } \; n $$.
If every row satisfies the Lindeberg condition follwing
$$ \sum_{j=1}^n\mathrm{E}|X_{n,j}|^2\mathbf{1}\{X_{n,j}>\varepsilon \} \rightarrow 0 \mbox{ as } \; n\rightarrow \infty $$
where, for all $\varepsilon > 0 $, then
$$ \dfrac{S_n}{s_n} →^d N(0, 1)\mbox{ as } \; n\to \infty $$
Am I right about it? Is there anything more I can say about this?