probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

, Dudley's theorem is a result relating the expected

upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an elem ...

and regularity properties of a

Gaussian process In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. ...

to its

entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...

and

covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...

structure.

History

The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by

Richard M. Dudley Richard Mansfield Dudley (July 28, 1938 – January 19, 2020) was Professor of Mathematics at the Massachusetts Institute of Technology. Education and career Dudley was born in Cleveland, Ohio. He earned his BA at Harvard College and received ...

. Dudley had earlier credited

Volker Strassen Volker Strassen (born April 29, 1936) is a German mathematician, a professor emeritus in the department of mathematics and statistics at the University of Konstanz. For important contributions to the analysis of algorithms he has received many aw ...

with making the connection between entropy and regularity.

Statement

Let (''X''_''t'')_{''t''∈''T''} be a Gaussian process and let ''d''_''X'' be the pseudometric on ''T'' defined by :

d_(s, t) = \sqrt. \,

For ''ε'' > 0, denote by ''N''(''T'', ''d''_''X''; ''ε'') the entropy number, i.e. the minimal number of (open) ''d''_''X''-balls of radius ''ε'' required to cover ''T''. Then :

\leq 24 \int_0^ \sqrt \, \mathrm \varepsilon.

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''_''X'').

References

* * {{ 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) Entropy Theorems regarding stochastic processes