Among the different generalizations of the CLT available on the web, I found these 


* CLT for the sum of non-identical (and independent) random variables
* CLT for the sum of identical (and independent) multivariate random variables.


However, I can't find any for the sum of non-identical (and independent) multivariate variables. 


Is it because it is straightforward ?
Would the final covariance matrix be simply the sum of the individual ones ?


Thanks



Multivariate Central Limit Theorem For Non-Identical Distribution
I always prefer to have error bounds for the CLT, so my favorite reference for your question is the paper "A Lyapunov type bound in $\mathbb{R}^d$" by Vidmantas Bentkus (Theory of Probability & Its Applications 49(2), 311--323, 2005).


From the abstract: Let $X\_1, \dots, X\_n$ be independent, mean-zero, $\mathbb{R}^d$-valued random variables. Let $S = X\_1 + \cdots + X\_n$ and let $C^2$ be the covariance matrix of $S$, assumed invertible. Let $Z$ be a $d$-dimensional Gaussian with mean zero and covariance $C^2$. Then for any convex subset $A \subseteq \mathbb{R}^d$, 


$$|\Pr[S \in A] - \Pr[Z \in A]| \leq O(d^{1/4}) \cdot \beta,$$
where $\beta = \sum\_{i} \mathbf{E}[|C^{-1}X\_i|^3]$. This is a $d$-dimensional generalization of the Berry--Esseen Theorem.


