Dudley's theorem

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.

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 (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).

Non-uniqueness

The solution is highly non-unique. There are infinitely many representation of the 0 function, and any of these can be added to a representation to obtain another representation.

References

• (See chapter 11)