Dudley's theorem explained

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.^[1] 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{ 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

E \left[ \sup _{t} X _{t} \right] \leq 24 \int ^{+inf ty} _{0} \sqrt{ log N(T, d _{X} ; \varepsilon)} 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

• 10.1016/0022-1236(67)90017-1. Dudley. Richard M.. Richard M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. Journal of Functional Analysis. 1. 1967. 3. 290 - 330. 0220340.
• Book: Ledoux. Michel. Talagrand . Michel . Michel Talagrand. Probability in Banach spaces. Springer-Verlag. Berlin. 1991. xii+480. 3-540-52013-9. 1102015. (See chapter 11)

Notes and References

1. V. N. Sudakov's work on expected suprema of Gaussian processes. High Dimensional Probability. VII. Christian. Houdré. David . Mason. Patricia. Reynaud-Bouret. Patricia Reynaud-Bouret. Jan. Jan Rosiński. Richard. Dudley. Richard M. Dudley. 2016. 37–43.