Asymptotic distribution of $\mathbb E_{\hat{P}_n}[Z]^T\operatorname{Cov}_{\hat{P}_n}[Z]^{-1}\mathbb E_{\hat{P}_n}[Z]$

Question

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.

Answer 1

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, 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.