Canonical pseudometric of a Gaussian process ($d(s,t)=(\mathbb{E}|X(s)-X(t){|}^2{)}^{1/2}$)

The canonical distance is the L2 norm of an increment of a Gaussian process. It is a pseudometric, since distinct parameters may represent equal random variables almost surely. Its metric covering number appears in the Dudley entropy integral. For standard fractional Brownian motion, $d(s,t)=|s-t{|}^H$; a normalization with twice the covariance function multiplies this distance by $\sqrt{2}$.