Dudley's entropy integral is a mathematical concept in the field of probability theory that describes a relationship involving the entropy of certain metric spaces and the concentration of measure phenomenon. It is named after the mathematician R. M. Dudley, who introduced the integral as part of his work on the uniform central limit theorem.
The Dudley's entropy integral is defined for a metric space equipped with a probability measure . Given a set and an -covering, the entropy of is the logarithm of the minimum number of balls of radius required to cover . Dudley's entropy integral is then given by the formula:
where is the covering number, i.e. the minimum number of balls of radius with respect to the metric that cover the space . [1]
Dudley's entropy integral arises in the context of empirical processes and Gaussian processes, where it is used to bound the supremum of a stochastic process. Its significance lies in providing a metric entropy measure to assess the complexity of a space with respect to a given probability distribution. More specifically, the expected supremum of a sub-gaussian process is bounded up to finite constants by the entropy integral. Additionally, function classes with a finite entropy integral satisfy a uniform central limit theorem. [2] [1]
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence, in the sense that the convergence is uniform over the domain. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every pointin. Described in an informal way, if converges to uniformly, then how quickly the functions approach is "uniform" throughout in the following sense: in order to guarantee that differs from by less than a chosen distance , we only need to make sure that is larger than or equal to a certain , which we can find without knowing the value of in advance. In other words, there exists a number that could depend on but is independent of , such that choosing will ensure that for all . In contrast, pointwise convergence of to merely guarantees that for any given in advance, we can find such that, for that particular, falls within of whenever .
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In probability theory, Donsker's theorem, named after Monroe D. Donsker, is a functional extension of the central limit theorem for empirical distribution functions. Specifically, the theorem states that an appropriately centered and scaled version of the empirical distribution function converges to a Gaussian process.
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In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
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In mathematics, a covering number is the number of balls of a given size needed to completely cover a given space, with possible overlaps between the balls. The covering number quantifies the size of a set and can be applied to general metric spaces. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric entropy, the number of points that fit in a space when constrained to lie at some fixed minimum distance apart.
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.
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