Let $\{(X_{n1},\ldots,X_{nk},\ldots),n=1,2,\ldots\}$ be a martingale difference sequence. In words we have $E[X_{nk}|\mathcal{F}_{n,k-1}] = 0$. If we know that $\sigma_k^2 = E[X_{nk}^2|\mathcal{F}_{n,k-1}] \in [\underline{\sigma}^2, \overline{\sigma}^2]$ where both $\sigma$'s are positive constants. Do we have, as $n\to\infty$,
$$
\frac{\sum (1-n^{-1})^k X_{nk}}{\sqrt{\sum (1-n^{-1})^{2k} \sigma_{nk}^2}} \Rightarrow N(0,1)
$$
and what will be the correct order of Berry Esseen bound?