# Sup-norm of Gaussian process

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.

• The answer would depend on the specific process. Take for example, $G_t$ to be 1) Brownian bridge process and 2) the standard Brownian motion. You get different results. Commented Sep 14, 2023 at 2:32
• In general, the supreme of Gaussian processes is a very hard to characterize, and has been the subject of intense study for the past few years. You might want to look into Talagrand's majorizing measure theorem tcsmath.wordpress.com/2010/07/18/…. Commented Sep 14, 2023 at 3:21
• @raghav can you show how the distribution of $\Vert G\Vert$ looks like in both cases? Commented Sep 15, 2023 at 3:34