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 $\beta\to 0^+$,
$$
\frac{\sum\_{k=1}^\infty (1-\beta)^k X\_{k}}{\sqrt{\sum\_{k=1}^\infty (1-\beta)^{2k} \sigma\_{k}^2}} \rightarrow N(0,1)
$$
in distribution, and what will be the correct order of Berry-Esseen bound?



Berry-Esseen bound for martingale sequence with varying and dependent variances
In the setting you describe, there's generally no CLT. Let me describe a counterexample showing that $\sum\_{i=1}^{k-1} X\_i / \sqrt{\sum\_{i=1}^{k-1} \sigma\_i}$ doesn't tend to normal. I'm pretty sure the same holds for the discounted averages as well.


The example is a random walk which is lazy when positive. More precisely, conditioned on $\mathcal{F\_{k-1}}$, if $\sum\_{i=1}^{k-1} X\_i <0$ then $X\_k$ is uniformly distributed in $\{-1,1\}$ and if $\sum\_{i=1}^{k-1} X\_i \ge 0$ then $X\_k$ is uniformly distributed in $\{-1,0,1\}$. The ratio between $P( \sum\_{i=1}^{k-1} X\_i <0)$ and $P(\sum\_{i=1}^{k-1} X\_i >0)$ tends to the ratio between the speed of the random walk when negative and when positive, which is 2:3, so the probability that the sum is positive tends to $\frac35$. I expect the same limit also for the discounted average.


