Assuming that for each fixed $k$, $(X\_{n,1},\ldots,X\_{n,k})\Longrightarrow(X\_1,\ldots,X\_k)$ where $X\_1,\ldots,X\_k$ are i.i.d. with mean zero and variance $\sigma^2$, will the array inherit the CLT from its limit? i.e. do I have (if $r\_n\to\infty$):
$\frac{1}{\sqrt{r\_n}}(X\_{n,1}+\cdots+X\_{n,r\_{n}})\Longrightarrow N(0,\sigma^2)$?



CLT for a converging array of random variables
No. Take any sequence $r\_n \to \infty$ and let $X\_{n,k}$ be iid $N(0,1)$ for $k \ne r\_n$, and $X\_{n,r\_n} = r\_n$. The hypothesis is satisfied because $(X\_{n,1}, \dots, X\_{n,k})$ are iid $N(0,1)$ as soon as $n$ is so large that $r\_n > k$. But
$$\frac{1}{\sqrt{r\_n}}(X\_{n,1} + \dots + X\_{n,r\_n}) \sim N(\sqrt{r\_n}, \frac{r\_n-1}{r\_n})$$
which is not even tight.


