Let $X_1,\dots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment.
If $d=1$, $\bar{X}_n=1/n\sum_{i=1}^nX_i$ and $S_n=1/(n−1)\sum_{i=1}^n(X_i−\bar{X}_n)^2$ then $$\sqrt{n}(\bar{X}_n−\mu)/S_n\rightarrow_nN(0,1)$$ in distribution. The same statement holds for $d>1$ if we additionally assume that we are given the covariance matrix $\Sigma$, that is $$\sqrt{n}(\bar{X}_n−\mu)\rightarrow_n\mathcal{N}(0,\Sigma).$$ and can be found in most standard references. However, I can't find a reference for the multivariate CLT with a covariance matrix estimator instead of $\Sigma$ as in the one-dimensional case $d=1$ given above (though I am pretty sure it must exists).