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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]] 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.
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| ==Statement of the theorem==
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| 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
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| :<math>d_{X}(s, t) = \sqrt{\mathbf{E} \big[ | X_{s} - X_{t} |^{2} ]}. \, </math>
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| 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
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| :<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> | |
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| 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>).
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| ==References==
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| * {{cite journal
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| | doi = 10.1016/0022-1236(67)90017-1
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| | last = Dudley
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| | first = Richard M.
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| | authorlink = Richard M. Dudley
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| | title = The sizes of compact subsets of Hilbert space and continuity of Gaussian processes
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| | journal = J. Functional Analysis
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| | volume = 1
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| | year = 1967
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| | pages = 290–330
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| | mr = 0220340
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| }}
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| * {{ cite book
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| | last1 = Ledoux
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| | first1 = Michel
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| | last2 = Talagrand | first2 = Michel | author2-link = Michel Talagrand
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| | title = Probability in Banach spaces
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| | publisher = Springer-Verlag
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| | location = Berlin
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| | year = 1991
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| | pages = xii+480
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| | isbn = 3-540-52013-9
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| | mr = 1102015
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| }} (See chapter 11)
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| [[Category:Entropy]]
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| [[Category:Probability theorems]]
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| [[Category:Stochastic processes]]
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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