Let $W\sim MVN(\mu, \Sigma)$, here $W$ and $\mu$ are $k\times 1$ vector and $\Sigma$ is $k\times k$ symmetric matrix. And the diagonal elements of $\Sigma$ are equal to one.
For this multivariate random variable, I want to find $$P(max_{k=1,\ldots,K} W_k^2 >q_\alpha)$$ where $q_\alpha$ is the upper $\alpha$ quantile for the distribution of $max^2_{k=1,\ldots,K}$.
Chi-square distribution or wishart distribution is applicable in this case? Is there any way to derive the probability analytically or numerically? Thanks in advance.