Английская Википедия:Dudley's theorem
Шаблон:Short description 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 first stated and proved by V. N. Sudakov, as pointed out in a paper by Richard M. Dudley.[1] Dudley had earlier credited Volker Strassen with making the connection between entropy and regularity.
Statement
Let (Xt)t∈T be a Gaussian process and let dX be the pseudometric on T defined by
- <math>d_{X}(s, t) = \sqrt{\mathbf{E} \big[ | X_{s} - X_{t} |^{2} ]}. \, </math>
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
- <math>\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.</math>
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
- Шаблон:Cite journal
- Шаблон:Cite book (See chapter 11)
