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, then do we have, as $n\to\infty$,
$$
\frac{\sqrt\beta\sum (1-\beta)^k X_{nk}}{\sqrt{\beta\sum (1-\beta)^{2k} \sigma_{nk}^2}} \Rightarrow N(0,1),
$$
and what will be the correct order of Berry Esseen bound?