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

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

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

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

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". Journal of 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)