I have a triangular array $(X_{i,n})$ that is not independent across $i$ for each $n$. But I can show that $\operatorname{Var}\left(n^{-1/2}\sum_{i = 1}^n X_{i,n}\right)$ is bounded and converges to $\Sigma$ as $n$ grows to infinity. I think that this is sufficient to establish Lindeberg's condition. Is it true?
For simplicity, I assume that $E(X_{i,n}) = 0$ for all $i$ and $n$.
Which central limit theorem can I apply on $n^{-1/2}\sum_{i = 1}^n X_{i,n}$? That is, I want to show that $\displaystyle n^{-1/2}\sum_{i = 1}^n X_{i,n} \to N(0, \Sigma)$.
No miracle can happen without any further assumption on the dependence across the rows $(X_{i,n})_{1\leqslant i\leqslant n}$. For example, let $X_{i,n}=Y_{i,n}-Y_{i-1,n}$, where $Y_{0,n}=0$, $\mathbb E[Y_{i,n}]=0$ and $\operatorname{Var}\left(n^{-1/2}Y_{n,n}\right)=1$. The central limit theorem would mean in this context that $n^{-1/2}Y_{n,n}$ in distribution to a standard normal random variable, which does not need to be the case, as we only required $Y_{n,n}$ to be centered and have variance $n$.