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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. The result was proved in a landmark 1967 paper of Richard M. Dudley; Dudley himself credited Volker Strassen for making the connection between entropy and regularity.

Statement of the theorem

Let (X_t)_t∈T be a Gaussian process and let d_X be the pseudometric on T defined by

$d_X(s,t)=\sqrt{\mathbf{E}\left[\right|X_s-X_t{|}^2]}.$

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

$\mathbf{E}\left[{\sup }_{t\in T}{X}_t\right]\le 24\underset{0}{\overset{+\mathrm{\infty }}{\int }}\sqrt{\log N(T,d_X;\epsilon )}\mathrm{d}\epsilon .$

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

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