Slepian's and Sudakov-Fernique comparison inequalities
/ SLEE-pee-an; soo-DAH-kov fer-NEEK /
Two Gaussian processes that you cannot compute directly can still be COMPARED if one is, coordinate by coordinate, 'more spread out' than the other. Slepian's inequality and its Sudakov-Fernique refinement make this precise: if you increase the gaps (the variance of differences) in a centered Gaussian process, you increase the expected supremum. They are the comparison tools that let you bound a complicated Gaussian maximum by a simpler one, and they are what makes the chaining / entropy program for suprema actually computable.
Let (X_t) and (Y_t) be centered Gaussian processes on a finite (or suitable) index set. Slepian's lemma: if E[X_s X_t] <= E[Y_s Y_t] for all s != t while E[X_t^2] = E[Y_t^2] for all t (equal variances, smaller off-diagonal correlations), then Y is stochastically larger in its supremum and ordering of events, e.g. P( max X_t <= a ) <= P( max Y_t <= a ); in particular E[max X_t] <= E[max Y_t]. Sudakov-Fernique drops the equal-variance requirement and compares directly through the incremental (canonical) metric: if E[(X_s - X_t)^2] <= E[(Y_s - Y_t)^2] for all s, t, then E[sup X_t] <= E[sup Y_t]. So the expected supremum is monotone in the size of the increment distances — a process whose points are 'farther apart' in the L^2 sense has a larger expected maximum. The intuition: more spread between coordinates means more chance for some coordinate to be large.
Why it matters: Sudakov-Fernique is the comparison half of the supremum theory. Combined with Borell-TIS (which handles fluctuations) it lets you bound E[sup X_t] for an unknown process by E[sup Y_t] for a model process you understand, and it underlies Sudakov's lower bound on the supremum via packing numbers, the dual partner of Dudley's upper bound. An honest caveat: Slepian's full event-ordering needs the equal-variance hypothesis; Sudakov-Fernique only delivers the inequality for the EXPECTED supremum (not full stochastic domination), and both genuinely require Gaussianity — they are false for general processes.
To bound E[max of a correlated Gaussian vector X], compare it to a vector Y of independent Gaussians with the SAME variances but larger increment distances (independence maximizes E[(Y_s - Y_t)^2] given variances). Sudakov-Fernique then gives E[max X] <= E[max Y] approximately sqrt(2 log n) sigma — positive correlations only shrink the expected maximum.
Larger L^2 increment distances mean a larger expected supremum; positive correlation reduces the maximum (Sudakov-Fernique).
Slepian's full ordering needs equal variances; Sudakov-Fernique relaxes that but only compares EXPECTED suprema, and both rely on Gaussianity — neither holds for general processes.