Dudley's theorem

Short description: Concept in probability theory

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 (X_t)_t∈T be a Gaussian process and let d_X be the pseudometric on T defined by

$d_X(s,t)=\sqrt{\mathbf{𝐄}[|X_s-X_t{|}^2]}.$

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

$\mathbf{𝐄}\left[\underset{t\in T}{\sup }{X}_t\right]\le 24\underset{0}{\overset{+\mathrm{\infty }}{\int }}\sqrt{\log N(T,d_X;\epsilon )}\mathrm{d}\epsilon .$

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_X).

References

• Dudley, Richard M. (1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". Journal of Functional Analysis 1 (3): 290–330. doi:10.1016/0022-1236(67)90017-1. 
• Ledoux, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9.  (See chapter 11)