Consider random walk in $\mathbb{Z}^d$, $d>1$, with $x(t) = x(t-1) + \xi$, where $\xi$ has some probability distribution in $\mathbb{Z}^d$ with finite support, expectation $m = \sum_{v \in {\mathbb{Z}^d}} P(\xi=v) v \, \, \in\mathbb{R}^d$.
What is the central limit theorem formulation for the variable $S_t = \frac{x(t) - mt}{\sqrt{t}}$? More precisely, how is distributed $S_{\infty}$? Is it simply the gaussian integrated with respect to every coordinate? How to define the variance for an probability distribution in $d>1$ dimensions?
The central limit theorem in multidimensional space is Theorem 3.9.6. (page 151) in the fourth edition of Probability: Theory and Examples by Richard Durrett. You can get a free copy of the book at http://www.math.duke.edu/~rtd/PTE/PTE4_1.pdf