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 of the theorem
Let (Xt)t∈T be a Gaussian process and let dX be the pseudometric on T defined by
![d_{{X}}(s,t)={\sqrt {{\mathbf {E}}{\big [}|X_{{s}}-X_{{t}}|^{{2}}]}}.\,](../I/m/770206df9ebe1f1c09dbea1684a2951cbae3309d.svg)
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
![{\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 .](../I/m/3cb0f5968f253567ccb5f75d5932eb0e68a73cd8.svg)
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. MR 0220340. doi:10.1016/0022-1236(67)90017-1.
- 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)