|
|
| Line 1: |
Line 1: |
| In [[probability theory]], '''Dudley’s theorem''' is a result relating the [[expected value|expected]] [[upper bound]] and [[Continuous stochastic process|regularity properties]] 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]] for making the connection between entropy and regularity.
| | Im addicted to my hobby Seashell Collecting. Seems boring? Not!<br>I also try to learn Danish in my free time.<br><br>my web-site; [http://www.handyladen24.net billige smartphones] |
| | |
| ==Statement of the theorem==
| |
| Let (''X''<sub>''t''</sub>)<sub>''t''∈''T''</sub> be a Gaussian process and let ''d''<sub>''X''</sub> 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'', ''d''<sub>''X''</sub>; ''ε'') the [[entropy number]], i.e. the minimal number of (open) ''d''<sub>''X''</sub>-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'', ''d''<sub>''X''</sub>).
| |
| | |
| ==References==
| |
| * {{cite journal
| |
| | doi = 10.1016/0022-1236(67)90017-1
| |
| | last = Dudley
| |
| | first = Richard M.
| |
| | authorlink = Richard M. Dudley
| |
| | title = The sizes of compact subsets of Hilbert space and continuity of Gaussian processes
| |
| | journal = J. Functional Analysis
| |
| | volume = 1
| |
| | year = 1967
| |
| | pages = 290–330
| |
| | mr = 0220340
| |
| }}
| |
| * {{ cite book
| |
| | last1 = Ledoux
| |
| | first1 = Michel
| |
| | last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand
| |
| | title = Probability in Banach spaces
| |
| | publisher = Springer-Verlag
| |
| | location = Berlin
| |
| | year = 1991
| |
| | pages = xii+480
| |
| | isbn = 3-540-52013-9
| |
| | mr = 1102015
| |
| }} (See chapter 11)
| |
| | |
| [[Category:Entropy]]
| |
| [[Category:Probability theorems]]
| |
| [[Category:Stochastic processes]]
| |
Im addicted to my hobby Seashell Collecting. Seems boring? Not!
I also try to learn Danish in my free time.
my web-site; billige smartphones