Let $f : \mathbb{R}^k \rightarrow \mathbb{R}$ and let $X$ be distributed k-dimensional normal with mean $0$ (with "arbitrary" covariance matrix). I am looking for references with bounds of the form: if $f \in \mathcal{F}$, then
$\sup_{t}P(|f(X) - t| \leq \epsilon) \leq ? $.
For example, results from Carbery and Wright (2001) provide useful very nice results if $f$ is polynomial. I think results from Ball (1993) can be applied here for certain convex functions but going this route would probably include in the rate a $k^{1/4}$ term that is too slow for my application.
Note: Apparently analogous statements to the Carbery and Wright (2001) results applied to the Gaussian setting above may hold if $f$ is a plurisubharmonic function as well. I'm still trying to digest this; perhaps an application of their results is among the more general statements available?
Note: I am being vague with my covariance matrix assumption and desire over $\mathcal{F}$ because this is for a statistics application where I am willing to put some restrictions on the covariance matrix but I do not want to have to assume it is the identity; $\mathcal{F}$ will hopefully encompass a class of statistics that are convex (with additional restrictions as needed).
