Weak convergence
Encyclopedia
In mathematics
, weak convergence may refer to:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, weak convergence may refer to:
- The weak convergence of random variablesConvergence of random variablesIn probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics and stochastic processes...
of a probability distributionProbability distributionIn probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
. - The weak convergence of a sequence of probability measures.
- The weak convergenceWeak convergence (Hilbert space)In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.-Definition:A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if...
of a sequence in a Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
, or, more generally, convergence in weak topologyWeak topologyIn mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
in a Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
or a topological vector spaceTopological vector spaceIn mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
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