Let $(G_t)_{t\in T}$ be a centered Gaussian process (with $T = [0,1]$). Can we say anything about the distribution of $$\Vert G\Vert := \sup_{t\in T}\vert G_t\vert?$$
For a multivariate normal (i.e., $T = \{1,2,...,n\}$), we know that the distribution function is given by $$F(x) = \int_{[- x, x]^n}\phi_\Sigma(u)\,\mathrm du,$$ where $\phi_\Sigma$ is the multivariate normal density of a centered $n$-variate normal distribution with covariance matrix $\Sigma$ (c.f., Distribution of the maximum of absolute value of multivariate Gaussian).
Still assuming $T = \{1,2,...,n\}$, it's also known that the expectation of $\Vert G\Vert$ can be bounded by $\sqrt{2\log(n)}$, which diverges as $n$ approaches infinity (despite a Gaussian process is known to be bounded almost surely?!).
I hope someone can help me sort my thoughts and resolve my confusion.