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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]] structureThe 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.
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==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''&nbsp;&gt;&nbsp;0, denote by ''N''(''T'',&nbsp;''d''<sub>''X''</sub>;&nbsp;''ε'') the [[entropy number]], i.e.&nbsp;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'',&nbsp;''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&ndash;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]]

Revision as of 19:33, 21 February 2014

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