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.
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.
Let (X<sub>t</sub>)<sub>tâÂÂT</sub> be a Gaussian process centered (with mean zero) and let d<sub>X</sub> be the pseudometric on T defined by
For õ > 0, denote by N(T, d<sub>X</sub>; õ) the entropy number, i.e. the minimal number of (open) d<sub>X</sub>-balls of radius õ required to cover T. Then
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, d<sub>X</sub>).