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This is the “weak convergence of laws without laws being defined” — except asymptotically. [1] In this case the term weak convergence is preferable (see weak convergence of measures), and we say that a sequence of random elements {X n} converges weakly to X (denoted as X n ⇒ X) if
In mathematics, strong convergence may refer to: The strong convergence of random variables of a probability distribution . The norm-convergence of a sequence in a Hilbert space (as opposed to weak convergence ).
This lists named topologies of uniform convergence. Compact-open topology. Loop space; Interlocking interval topology; Modes of convergence (annotated index) Operator topologies; Pointwise convergence. Weak convergence (Hilbert space) Weak* topology; Polar topology; Strong dual topology; Topologies on spaces of linear maps
In analysis, a topology is called strong if it has many open sets and weak if it has few open sets, so that the corresponding modes of convergence are, respectively, strong and weak. (In topology proper, these terms can suggest the opposite meaning, so strong and weak are replaced with, respectively, fine and coarse.)
The definition of weak convergence can be extended to Banach spaces. A sequence of points ( x n ) {\displaystyle (x_{n})} in a Banach space B is said to converge weakly to a point x in B if f ( x n ) → f ( x ) {\displaystyle f(x_{n})\to f(x)} for any bounded linear functional f {\displaystyle f} defined on B {\displaystyle B} , that is, for ...
In mathematics, weak convergence may refer to: Weak convergence of random variables of a probability distribution; Weak convergence of measures, of a sequence of probability measures; Weak convergence (Hilbert space) of a sequence in a Hilbert space more generally, convergence in weak topology in a Banach space or a topological vector space
For (,) a measurable space, a sequence μ n is said to converge setwise to a limit μ if = ()for every set .. Typical arrow notations are and .. For example, as a consequence of the Riemann–Lebesgue lemma, the sequence μ n of measures on the interval [−1, 1] given by μ n (dx) = (1 + sin(nx))dx converges setwise to Lebesgue measure, but it does not converge in total variation.
Both the weak topology and the weak* topology are special cases of a more general construction for pairings, which we now describe.The benefit of this more general construction is that any definition or result proved for it applies to both the weak topology and the weak* topology, thereby making redundant the need for many definitions, theorem statements, and proofs.