Dudley's theorem

Last updated

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.

Contents

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)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

  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. doi:10.1007/978-3-319-40519-3_2.