Let $z$ be the standard gaussian vector, $z\sim N(0,I_k)$. I was wondering if one can find a lower bound for the following expression $$1-P\{\Sigma^{\frac{1}{2}}z\leq t\}$$
where $t\in \mathbb{R}^k$ and $\Sigma^{\frac{1}{2}}z\sim N(0,\Sigma)$.
By using Slepian's inequality one can find an upper bound by simplifying the covariance matrix. I don't know how we can find a lower bound for the above aexpression.