If $X_1, X_2, X_3, ...$ are random variables and $(\bar{X_n}, \frac{1}{n} \sum_{1}^{n} X_i^2 - (\bar{X_n})^2)$ has some asymptotic distribution, will $(\bar{X_n}, \frac{1}{n-1} \sum_{1}^{n} X_i^2 - \frac{n}{n-1}(\bar{X_n})^2)$ have the same asymptotic distribution (because for large $n$, $\frac{1}{n} \approx \frac{1}{n-1}$ and $\frac{n}{n-1} \approx 1$)? Or do we also need to assume the $X_i$'s are also i.i.d?
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2https://en.wikipedia.org/wiki/Slutsky%27s_theorem – Christoph Hanck Oct 10 '22 at 09:24