I observe a set/realisation of $n$ i.i.d. $\{X_1, X_2, ..., X_n\}$. Because of the Central Limit Theorem, I know that repeating such an observation enough times, the pdf of the mean of such $n$ samples, $\bar{X}_n$, has variance $\sigma^2/n$, with $\sigma^2$ the original variance of $X$.
Now I instead take the mean $$\bar{X}_{n,m}=\frac{1}{m} \sum_{i=n-m}^n \bar{X}_i$$
Is it true that the variance of the pdf of $\bar{X}_{n,m}$ scales as $\sigma^2/(n/m)$?