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Dudley's theorem

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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.

Contents

History

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.

Statement of the theorem

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 ] .

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 .

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

Dudley's theorem Wikipedia
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