- 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. The result was proved in a landmark 1967 paper of Richard M. Dudley; Dudley himself credited Volker Strassen for making the connection between entropy and regularity.
Statement of the theorem
Let (Xt)t∈T be a Gaussian process and let dX be the pseudometric on T defined by
For ε > 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
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).
References
- Dudley, Richard M. (1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". J. Functional Analysis 1: 290–330. doi:10.1016/0022-1236(67)90017-1. MR0220340.
- Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR1102015. (See chapter 11)
Categories:- Entropy
- Probability theorems
- Stochastic processes
Wikimedia Foundation. 2010.