# Does the multivariate Central Limit Theorem (CLT) hold when variables exhibit perfect contemporaneous dependence?

The title sums up my question, but for clarity consider the following simple example. Let $X_i \overset{iid}{\backsim} \mathcal{N}(0, 1)$, $i = 1, ..., n$. Define: $$S_n = \frac{1}{n} \sum_{i=1}^n X_i$$ and $$T_n = \frac{1}{n} \sum_{i=1}^n (X_i^2 - 1)$$ My question: Even though $S_n$ and $T_n$ are perfectly dependent when $n = 1$, do $\sqrt{n} S_n$ and $\sqrt{n} T_n$ converge to a joint normal distribution as $n \rightarrow \infty$?

The motivation: My motivation for the question stems from the fact that it feels odd (but wonderful) that $S_n$ and $T_n$ are perfectly dependent when $n = 1$, yet the implication of the multivariate CLT is that they approach independence as $n \rightarrow \infty$ (this would follow since $S_n$ and $T_n$ are uncorrelated for all $n$, hence if they are asymptotically joint normal, then they must also be asymptotically independent).

ps, If you can provide any references etc then all the better!

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 No answer, but a comment. I don't find this very surprising. The dependence you note for n = 1 quickly decreases as n goes up. – Erik Nov 19 '12 at 11:25 @egbutter has provided a fine answer. If you are still looking for some alternative or some additional intuition, ping me and I will see about writing up something a little bit different. – cardinal Nov 29 '12 at 1:04 @cardinal Thanks very much for the offer, but I'm fairly happy at this point - I awarded the bounty to egbutter. I think I've got the intuition. My main purpose in posting was to see if someone jumped in and said "No no no you've got it all wrong because of..." :-) Cheers. – Colin T Bowers Nov 29 '12 at 3:52