Let $\{(X_{n,1},\ldots,X_{n,k},\ldots),n=1,2,\ldots\}$ be a triangular array of martingale difference sequences. In words we have $E[X_{n,k}|\mathcal{F}_{n,k-1}] = 0$, where $\mathcal{F}_{n,k-1}$ is the information by time $k-1$. If we know that $\sigma_{n,k}^2 = E[X_{n,k}^2|\mathcal{F}_{n,k-1}] \in [\underline{\sigma}^2, \overline{\sigma}^2]$ (here note that $\sigma_{n,k}^2$ is a random variable measurable w.r.t. $\mathcal{F}_{n,k-1}$), where both $\underline{\sigma}$ and $\overline{\sigma}$ are positive constants, then do we have, as $n\to\infty$,
$$
\frac{\sum (1-n^{-1})^k X_{n,k}}{\sqrt{\sum (1-n^{-1})^{2k} \sigma_{n,k}^2}} \rightarrow N(0,1),
$$
and what will be the correct order of Berry-Esseen bound?