If $X\_i$ is a sequence of $d$ dimensional i.i.d. integer valued random vectors with covariance matrix $\Sigma$ and $\mathbb{E}(X\_i) = \mu$. Let each element of $X\_i$ be chosen i.u.d. from $\{-1,1\}$. We know from the multidimensional central limit theorem that


$$ \frac{1}{\sqrt{n}}\sum\_{i=1}^n (X\_i - \mu)\ \stackrel{D}{\rightarrow}\ \mathcal{N}\_d(0,\Sigma).$$


There are in fact a whole series of multivariate central limit theorems that weaken the requirements for dependence and/or that the summands must be identical. See for example [Theorems D.18A](http://people.stern.nyu.edu/wgreene/Text/Edition7/Z01_GREE5381_07_SE_APP_D-E-checked.pdf) onwards.


These convergence results appear all to require that the dimension size $d$ is fixed.



> 
> What happens if the dimension size $d$ grows as a function of the number of terms in the sum? For
>  example, say $d = \lfloor \sqrt{n} \rfloor$ or $d = \lfloor
> > \frac{n}{\ln n}\rfloor$. Are there results which say under what
>  conditions $\frac{1}{\sqrt{n}}\sum\_{i=1}^n (X\_i - \mu)$ converges to
>  a multivariate Gaussian?
> 
> 
> 


To be clear, for a fixed $n$ all the vectors in the sum will have the same dimension $d$. However, that dimension will be a function of $n$ and my assumption is that for $d=n$, say, then it may be hard to argue that $ \frac{1}{\sqrt{n}}\sum\_{i=1}^n (X\_i - \mu)$ is well approximated by $\mathcal{N}\_n(0,\Sigma)$.



Multivariate CLT with varying dimension size
I interpret this as being about (Berry--Esseen-style) closeness of $X\_1 + \cdots + X\_n$ to the $d$-dimensional Gaussian distribution, when $n$ is fixed and when the dependence on the dimension $d$ is carefully taken into account.


It depends on what class of 'tests' you use to measure closeness, but the best general result along these lines I know is from Bentkus, "A Lyapunov type bound in Rd":


<http://epubs.siam.org/doi/abs/10.1137/S0040585X97981123>


He shows closeness for all convex test sets, with the dependence on the dimension being $d^{1/4}$.


