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.

Partial solution

By the Cramér-Rao bound, one has $\operatorname{Cov}_{\hat{P}_n}[Z] \succeq 1$ and so $$ R_n \le \|\mathbb E_{\hat{P}_n}[Z]\|_2^2. $$ Please correct me if I'm wrong.

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.

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.

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.

Partial solution

By the Cramér-Rao bound, one has $\operatorname{Cov}_{\hat{P}_n}[Z] \succeq 1$ and so $$ R_n \le \|\mathbb E_{\hat{P}_n}[Z]\|_2^2. $$ Please correct me if I'm wrong.

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.

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.

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.