Let $g = (g_1, \dots, g_d)$ be a sequence of independent standard Normal random variables, and suppose $\Sigma$ is a $d \times d$ (deterministic), real, symmetric, positive definite matrix. The Hanson-Wright inequality ensures that for $t > 0$,the upper and lower tails have the bound $$ \max\bigg(\mathbb{P}\Big\{g^T \Sigma^{-1} g - \mathrm{Tr}(\Sigma^{-1}) \geq t\Big\}, \mathbb{P}\Big\{g^T \Sigma^{-1} g - \mathrm{Tr}(\Sigma^{-1}) \leq - t\Big\}\bigg) \leq \exp\Big(-c \min\big\{\frac{t^2}{\|\Sigma^{-1}\|_F^2}, \frac{t}{\|\Sigma^{-1}\|_{\rm op}} \big\}\Big). $$ Above, $c > 0$ is a universal constant.
My question: do similar guarantees hold when $\Sigma$ is a random matrix? I am particularly interested in the case when we have $$ \Sigma = \frac{1}{n} \sum_{j=1}^n x_j \otimes x_j. $$ (We assume that $x_j \in \mathbb{R}^d$ are iid, and have law such that when $n > d$, $\Sigma$ is nonsingular almost surely.)
In this case, also assuming $x_j$ are mutually independent of $g$, I am wondering under what conditions we can expect to have exponential tail bounds for $$ \mathbb{P}_{x, g}\Big\{g^T \Sigma^{-1} g - \mathbb{E}_x[\mathrm{Tr}(\Sigma^{-1})] \geq t\Big\} \quad \mbox{or} \quad \mathbb{P}_{x,g}\Big\{g^T \Sigma^{-1} g - \mathbb{E}_x[\mathrm{Tr}(\Sigma^{-1})] \leq - t\Big\}, $$ where the probability is now over $g_1, \dots, g_d, x_1, \dots, x_n$. Any references or pointers to existing results are greatly appreciated.
