Let the vector $\boldsymbol x$ be a draw of $n$ values from a multivariate normal distribution with zero mean. $$ \boldsymbol x \sim \mathcal{N}(\boldsymbol 0, \Sigma) $$ It may be assumed that $\Sigma$ is standardised.
Let the scalar $y$ be the variance of $\boldsymbol x$: $$ y = var(\boldsymbol x) $$ Then, what is the distribution of $y$, given $n$ and $\Sigma$? $$ y \sim ?(n, \Sigma) $$
Context: an autoregressive process (AR1) generates time series where values close in time are more correlated than values further apart. Generating $n$ such values is equivalent to sampling from a multivariate normal distribution with a particular covariance structure, namely $\Sigma_{ij} = \rho^{|t_i - t_j|}$. What is the variance $y$ of the values of these time series, and how is that distributed when multiple time series are generated?