# Multivariate Central Limit Theorem For Non-Identical Distribution [closed]

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

-

## closed as off-topic by Did, Ryan Budney, Mark Meckes, Andres Caicedo, Chris GodsilJul 15 '13 at 3:08

• This question does not appear to be about research level mathematics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

## 1 Answer

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.

-
I kind of agree with the (now-deleted) comment made by OP that it's not clear why this question is On Hold. It's not so easy to find clear statements about the multidimensional, non-iid, CLT in the literature. For example, the (first?) paper on the multidim CLT by Sazonov only treats the iid case, and the standard textbook by Bhattacharya and Rao is a bit of a nightmare to read (in my opinion). –  Ryan O'Donnell Jul 18 '13 at 20:51