I am wondering if there is a closed-form expression for the probability that a Gaussian random vector $\boldsymbol{X}$ falls in-between some bounds as specified by a different Gaussian random variable $\boldsymbol{Y}$.
Specifically, let $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}^{x}, \boldsymbol{\Sigma}^{x})$ and $\boldsymbol{Y} \sim \mathcal{N}(\boldsymbol{\mu}^{y}, \boldsymbol{\Sigma}^{y})$ (and $n_{x} = n_{y}$, i.e. same dimensionality) be independent random vectors. Let $\eta > 0$ be some bound and define the ellipses $\mathcal{E}^{y} = [ \boldsymbol{x}^{\top} \boldsymbol{\Sigma}^{y} \boldsymbol{x} \leq \eta ]$ and $\mathcal{E}^{x} = [ \boldsymbol{x}^{\top} \boldsymbol{\Sigma}^{x} \boldsymbol{x} \leq \eta ]$. Its easy to see that $\mathbb{P}[Y \in \mathcal{E}^{y}]$ can be directly calculated from a $\chi^{2}$ distribution with $n_{y} = n_{x}$ degrees of freedom (and likewise $\mathbb{P}[X \in \mathcal{E}^{x}]$). However, I'm interested in the following:
"Is there a closed-form expression for the probability $\mathbb{P}[X \in \mathcal{E}^{y}]$?"
I have tried computing this probability using both numerical integration and Monte Carlo methods, but I was wondering if there would be something more sophisticated. I also tried to look at the Kullback-Leibler divergence but couldn't find a direct link to probability. Any help would be much appreciated, thanks!
P.S.: Sorry for the weird notation, MathJax does seem to render when writing subscripts and curly braces.