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 Dudley, "V. N. Sudakov's work on expected suprema of Gaussian processes," in High Dimensional Probability VII, Eds. C. Houdré, D. M. Mason, P. Reynaud-Bouret, and Jan Rosiński, Birkhăuser, Springer, Progress in Probability 71, 2016, pp. 37–43. Dudley had earlier credited Volker Strassen with making the connection between entropy and regularity.Statement
Let t∈T be a Gaussian process and let dX be the pseudometric on T defined by
For ε >; 0, denote by N the entropy number, i.e. the minimal number of dX-balls of radius ε 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 continuous on.
