Conditions for convergence and the multivariate central limit theorem Consider $r$ independent but not identically distributed random vectors $x_1,\dots,x_r$, each of dimension $d$. The elements of $x_i$ are integer valued elements in the range $-d,\dots,d$ with mean $0$ and variance $d$. Each element of each vector $x_i$ is in fact just distributed as a simple symmetric random walk with $d$ steps. The elements of $x_i$ may not however be independent. > Under what conditions does $\sum_{j=1}^r x_i$ (suitably scaled) converge to a multivariate Gaussian as $r$ and $d$ go to infinity? Is there some relationship between $d$ and $r$ that is necessary for convergence?

A sufficient condition, given that $x_i$ are bounded, is that $\frac{1}{n}\sum_{i=1}^n C_i \to C$, where $C_i$ are the covariance matrices. Then $\frac{1}{\sqrt{n}}( \sum x_i - \mu_i)\to N(0,C)$. See eg here.