If $\mu\ne0$, then the distribution of $R_n$ is asymptotically normal with asymptotic mean $\mu^T\Sigma^{-1}\mu$ and an explicit asymptotic variance $\tilde\sigma/\sqrt n$; see e.g. [Theorem 3.9, page 1018][1], where a bound on the rate of convergence is also given. More specifically, $\tilde\sigma=\sqrt {{\tilde\sigma}^2}$ and 
\begin{equation}
	\tilde\sigma^2=EL(V)^2=E(2\xi-\xi^2+\mu^T\Sigma^{-1}\mu)^2,
\end{equation}
where 
\begin{equation}
	V:=(Y-EY,(Y-EY)(Y-EY)^T-I),\quad Y:=\Sigma^{-1/2}z_1, 
\end{equation}
\begin{equation}
	L(x_1,x_2):=2x_1^T\,EY-EY^T\,x_2\,EY,
\end{equation}
\begin{equation}
	\xi:=EY^T\,(Y-EY)=\mu^T\Sigma^{-1}(z_1-\mu). 
\end{equation}

If $\mu=0$, then the distribution of properly normalized $R_n$ is asymptotically chi-squared; see e.g. [Theorem 3, page 48][2].


  [1]: https://projecteuclid.org/euclid.ejs/1460463653
  
  [2]: https://www.sciencedirect.com/science/article/pii/S0047259X84710128