Dudley's entropy bound
/ DUD-lee /
Borell-TIS tells you the supremum of a Gaussian process fluctuates by at most sigma; the genuinely hard quantity is the MEAN of that supremum. Dudley's entropy bound is the standard upper estimate for it: it controls E[sup X_t] by a single integral of the metric entropy of the index set, measured in the natural distance that the process itself induces. It is the payoff of the chaining technique — successively refining a net over the index set.
Equip the index set T with the canonical (intrinsic) metric d(s, t) = sqrt( E[(X_s - X_t)^2] ), the L^2 distance between the process values. Let N(T, d, epsilon) be the covering number — the fewest balls of radius epsilon needed to cover T — and call log N(T, d, epsilon) the metric entropy. Dudley's bound states E[ sup_(t in T) X_t ] <= C times integral_0^infinity sqrt( log N(T, d, epsilon) ) d epsilon, for a universal constant C (the integral effectively runs up to the diameter of T). The proof is chaining: approximate each t by its nearest point in a sequence of finer and finer epsilon-nets, write X_t as a telescoping sum of increments between consecutive net levels, bound each level's contribution using the Gaussian (sub-Gaussian) tail of increments at scale epsilon, and sum the geometric ladder — the sqrt(log N) is exactly the maximum of finitely many sub-Gaussians at each scale. Sudakov's inequality gives a matching LOWER bound, sup_epsilon epsilon sqrt(log N(T, d, epsilon)) <= C' E[sup X_t], so entropy controls the supremum from both sides up to constants.
Why it matters: Dudley turns a probabilistic quantity into a geometric/combinatorial one — count epsilon-balls — and is the basic tool for proving sample-path continuity and boundedness of Gaussian processes and for bounding empirical processes in statistics and learning theory. The honest caveat is that Dudley is NOT tight in general: there exist Gaussian processes whose true E[sup] is strictly smaller than the entropy integral. The sharp, two-sided characterization is Talagrand's generic chaining / majorizing measures (the gamma_2 functional), of which Dudley's integral is a clean but lossy upper bound.
For Brownian motion on [0, 1], the canonical distance is d(s, t) = sqrt( E[(B_s - B_t)^2] ) = sqrt(|s - t|). Covering [0, 1] to resolution epsilon in this metric needs about 1/epsilon^2 balls, so log N approximately 2 log(1/epsilon), and integral_0^1 sqrt(2 log(1/epsilon)) d epsilon converges — giving E[sup_(t<=1) B_t] < infinity, i.e. boundedness, directly from the entropy integral.
Dudley bounds the expected supremum by the entropy integral in the process's own L^2 metric; a finite integral proves boundedness.
Dudley's integral is an upper bound only and can be loose; the sharp two-sided answer is Talagrand's generic chaining (the gamma_2 functional), with Sudakov giving a matching lower bound.