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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).

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    $\begingroup$ Halbert White's "asymptotic theory for econometricians" should have it. Basically, by slutsky's theorem, if you have a consistent estimator, $\hat{\Sigma}$, then the proof of the estimated case follows the same reasoning. But you're better off checking White's text than having me try to explain it. $\endgroup$ – mark leeds Jul 27 '15 at 16:31
  • $\begingroup$ Thanks! I thought it'd boil down to Slutsky but I wasn't sure about sufficient conditions for consistent estimators...the book has a whole chapter about it. $\endgroup$ – john Jul 29 '15 at 15:24
  • $\begingroup$ That's a really nice book that doesn't get enough hype. And since White passed away, it will probably now get even less. IMHO, if you have hamilton's text and white's text, you pretty much have the whole "kit and caboodle" in econometric time series. All the best. $\endgroup$ – mark leeds Jul 30 '15 at 4:29

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