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 (Xt)tT be a Gaussian process and let dX be the pseudometric on T defined by

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

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". 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)
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