Dudley's theorem

In probability theory, Dudley's theorem is a result relating the expected upper bound and regularity properties of a Gaussian process to its entropy and covariance structure.

History
The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by Richard M. Dudley. Dudley had earlier credited Volker Strassen with making the connection between entropy and regularity.

Statement
Let (Xt)t∈T be a Gaussian process and let dX be the pseudometric on T defined by


 * $$d_{X}(s, t) = \sqrt{\mathbf{E} \big[ | X_{s} - X_{t} |^{2} ]}. \, $$

For ε &gt; 0, denote by N(T, dX; ε) the entropy number, i.e. the minimal number of (open) dX-balls of radius ε required to cover T. Then


 * $$\mathbf{E} \left[ \sup_{t \in T} X_{t} \right] \leq 24 \int_0^{+\infty} \sqrt{\log N(T, d_{X}; \varepsilon)} \, \mathrm{d} \varepsilon.$$

Furthermore, if the entropy integral on the right-hand side converges, then X has a version with almost all sample path bounded and (uniformly) continuous on (T, dX).