Under very general conditions on the random $p$-dimensional vector $Z$, what can be said about the asymptotic distribution of the (random) scalar quantity $R\_n := \mathbb E\_{\hat{P}\_n}[Z]^T\operatorname{Cov}\_{\hat{P}\_n}[Z]^{-1}\mathbb E\_{\hat{P}\_n}[Z]$ ?


Here, $\mathbb E\_{\hat{P}\_n}[Z] = (1/n)\sum\_{i=1}^nz\_i \in \mathbb R^p$ is the empirical mean of $Z$ from an i.i.d sample $z\_1,\ldots,z\_n$, and $\operatorname{Cov}\_{\hat{P}\_n}[Z] \in \mathbb R^{p \times p}$ is the empirical covariance matrix.


Notes
=====


My ultimate goal is to understand the rate of growth of $R\_n$ as a function of $n$.


Observations
============


My wild guess is that $R\_n$ should be "concentrated" around $\mu^T\Sigma^{-1}\mu$ where $\mu \in \mathbb R^p$ is the mean of $Z$ and $\Sigma \in \mathbb R^{p \times p}$ is its covariance matrix.



Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$
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 $\tilde\sigma^2/n$; see e.g. [Theorem 3.9, page 1018](https://projecteuclid.org/euclid.ejs/1460463653), where a bound on the rate of convergence is also given. More specifically, 
\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](https://www.sciencedirect.com/science/article/pii/S0047259X84710128).


