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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

$d_{X}(s,t)={\sqrt {\mathbf {E} {\big [}|X_{s}-X_{t}|^{2}]}}.\,$

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

$\mathbf {E} \left[\sup _{t\in T}X_{t}\right]\leq 24\int _{0}^{+\infty }{\sqrt {\log N(T,d_{X};\varepsilon )}}\,\mathrm {d} \varepsilon .$

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, d_X).

References

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