Dudley entropy integral

Let $(X_t{)}_{t\in T}$ be a centered Gaussian process with a separable version: its relevant suprema are captured by a countable dense index subset. Give $T$ the canonical pseudometric of a Gaussian process $d(s,t)=\Vert X_s-X_t{\Vert }_2$, and assume it is totally bounded. With metric covering number $N(\epsilon ,T,d)$ and diameter $D$, the Dudley entropy integral boundsfor a universal constant $C$. Finiteness of the integral gives a finite expected supremum. For an absolute supremum, include an anchor: ${\sup }_t|X_t|\le |X_{t_0}|+{\sup }_t|X_t-X_{t_0}|$. The anchor term cannot generally be omitted.

The proof builds successively finer nets. Write each increment as a telescoping sum between neighboring net scales, bound each finite Gaussian maximum by its scale times the square root of the logarithmic net size, and sum over scales. This becomes the displayed entropy integral.^[ref]

The version assumption is substantive. If $U$ is uniform on $[0,1]$ and $X_t=1_{\{U=t\}}$, every finite vector is zero almost surely and is therefore a degenerate Gaussian vector; the canonical pseudometric is identically zero. Yet ${\sup }_t{X}_t=1$ for every outcome. Its all-zero separable modification has supremum zero. Finite-dimensional laws alone do not control the supremum of an arbitrary uncountable version.