Let X be an $\mathbb R^d$-valued random variable with distribution $N_d(0,\Sigma)$. I'm looking for a function $f$ such that $$P(|X_1|\leq M, |X_2|\leq M,\dots, |X_d|\leq M)\geq f(M),$$ and such that $f(M)\to 1$ as $M\to\infty$ (i.e. giving the convergence rate of that probability to 1). Of course $f$ will somehow depend on $\Sigma$ (most certainly on its rank, maybe on its norm defined in some way). I tried to connect it to the trivial case $\Sigma=I_d$ but with no success so far. Do you have any ideas?

EDIT: of course an idea is to use $x'\Sigma^{-1}x\leq |x|^2/\lambda_1$ (where $\lambda_1$ is the lowest eigenvalue of $\Sigma$) and then use a bound on the standard normal. But again this requires invertibility of $\Sigma$, and I don't want to assume that. When the rank of $\Sigma$ is lower than $d$, another idea I had was to write the last $n-d$ components of $X$ as linear combinations of the other ones, but I'm still unable to simplify the expressions enough to obtain simple convergence rates in terms of some basic properties of $\Sigma$.