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 proved in a landmark 1967 paper of Richard M. Dudley; Dudley himself credited Volker Strassen with making the connection between entropy and regularity.

Statement of the theorem

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

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

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

${\mathbf {E}}\left[\sup _{{t\in T}}X_{{t}}\right]\leq 24\int _{0}^{{+\infty }}{\sqrt {\log N(T,d_{{X}};\varepsilon )}}\,{\mathrm {d}}\varepsilon .$

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, d_X).

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. MR 0220340.
• Ledoux, Michel; Talagrand, Michel (1991). Probability in Banach spaces. Berlin: Springer-Verlag. pp. xii+480. ISBN 3-540-52013-9. MR 1102015. (See chapter 11)