Английская Википедия:Dudley's theorem

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

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:Reflist

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  1. ↑ Шаблон:Cite conference
Источник — http://wikihandbk.com/ruwiki/index.php?title=Английская_Википедия:Dudley%27s_theorem&oldid=14396851