Dudley's theorem

Concept in probability theory

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

d X ( s , t ) = E [ | X s − X t | 2 ] . {\displaystyle 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

E [ sup t ∈ T X t ] ≤ 24 ∫ 0 + ∞ log ⁡ N ( T , d X ; ε ) d ε . {\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, dX).

References

  1. ^ Dudley, Richard (2016). Houdré, Christian; Mason, David; Reynaud-Bouret, Patricia; Jan Rosiński, Jan (eds.). V. N. Sudakov's work on expected suprema of Gaussian processes. High Dimensional Probability. Vol. VII. pp. 37–43.
  • Dudley, Richard M. (1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". Journal of Functional Analysis. 1 (3): 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)