Let $(X_{1},\ldots,X_{k},\ldots)$ be a martingale difference sequence, i.e. $$ E[X_{k}|\mathcal{F}_{k-1}] = 0 $$ where $\mathcal{F}_{k-1}$ is the $\sigma$-algebra filtration at $k-1$.

Let $\sigma_{k}^2 = E[X_{k}^2|\mathcal{F}_{k-1}]$. Here note that $\sigma_{k}^2$ is a random variable measurable w.r.t.~$\mathcal{F}_{k-1}$. If we know that $\sigma_{k}^2 \in [\underline{\sigma}^2, \overline{\sigma}^2]$, where both $\underline{\sigma}$ and $\overline{\sigma}$ are positive constants, then do we have, as $\beta\to 0^+$, $$ \frac{\sum_{k=1}^\infty (1-\beta)^k X_{k}}{\sqrt{\sum_{k=1}^\infty (1-\beta)^{2k} \sigma_{k}^2}} \Rightarrow N(0,1), $$ and what will be the correct order of Berry-Esseen bound?