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)t∈T be a Gaussian process and let dX be the pseudometric on T defined by
\( d_{{X}}(s,t)={\sqrt {{\mathbf {E}}{\big [}|X_{{s}}-X_{{t}}|^{{2}}]}}.\, \)
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
\( {\mathbf {E}}\left[\sup _{{t\in T}}X_{{t}}\right]\leq 24\int _{0}^{{+\infty }}{\sqrt {\log N(T,d_{{X}};\varepsilon )}}\,{\mathrm {d}}\varepsilon . \)
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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