Dudley's theorem

In probability theory, Dudley’s theorem is a result relating the expected upper bound and 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 with making the connection between entropy and regularity.

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

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

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).

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