If $\mu\ne0$, then the distribution of $R_n$ is asymptotically normal with asymptotic mean $\mu^T\Sigma^{-1}\mu$ and a certain explicit variance; see e.g. Theorem 3.9, page 1018, where a bound on the rate of convergence is also given.
If $\mu=0$, then the distribution of properly normalized $R_n$ is asymptotically chi-squared; see e.g. Theorem 3, page 48.