Let $M,N\to\infty$ with $M/N\to 0$. Suppose there is a random sequence $X_{mi}$ with mean 0 and variance $\sigma_{m}^{2}$. $X_{mi}$ is independent of $X_{mj}$ for all $i\neq j$ but $X_{mi}$ is correlated with $X_{m'j}$ for any $m\neq m'$ and all $i,j$.
My question is whether the following is true. $$ \frac{1}{\sqrt{N}}\sum_{i=1}^{N}\left(\frac{\sum_{m=1}^{M}X_{mi}}{\sqrt{Var(\sum_{m=1}^{M}X_{mi})}}\right)\overset{d}{\to}N(0,1). $$
I know this holds by central limit theorem if $M$ is fixed. However, now that $M\to\infty$, I'm not quite sure whether I can still treat the whole thing in the parentheses as a random variable with mean 0 and unit variance, or if I need to think about $\sum_{i}X_{mi}/\sqrt{N}$ as an empirical process indexed by $m$ and show it's weakly converging to some Gaussian process.
