Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal.

What do we know about the mean and variance of this asymptotic Gaussian? I know that if $x_i \sim \mathcal{N}(\mu,\sigma)$ that the sample mean has distribution $\mathcal{N}(\mu, \sqrt{\sigma}/n)$. What about if $x_i$ comes from some arbitrary distribution (say, exponential)?

Any references would be appreciated as well.

Let $\{x_i\}$ be a set of iid random variables (not necessarily Gaussian distributed). The CLT states that $\frac{1}{n}\sum_{i=1}^n x_i$ is asymptotically normal.

What do we know about the mean and variance of this asymptotic Gaussian? I know that if $x_i \sim \mathcal{N}(\mu,\sigma)$ that the sample mean has distribution $\mathcal{N}(\mu, \sqrt{\sigma}/n)$. What would the approach be if $x_i$ comes from some arbitrary distribution (say, exponential)?

And in the multivariate case (i.e. $x_i \in \mathbb{R}^n$) or more generally the continuous case (i.e. $x_i(t)$ is a function over a continuous-valued $t$), we would also need to find covariance.