Return to Question

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?

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?

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?

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?

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 $\sigma$'s 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?

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?