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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, 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
This article is supplemental for “Convergence of random variables” and provides proofs for selected results. Several results will be established using the portmanteau lemma: A sequence {X n} converges in distribution to X if and only if any of the following conditions are met:
If X n: Ω → X is a sequence of random variables then X n is said to converge weakly (or in distribution or in law) to the random variable X: Ω → X as n → ∞ if the sequence of pushforward measures (X n) ∗ (P) converges weakly to X ∗ (P) in the sense of weak convergence of measures on X, as defined above.
In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probability space.
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also ...
In 1949 Doob asked whether the convergence in distribution held for more general functionals, thus formulating a problem of weak convergence of random functions in a suitable function space. [3] In 1952 Donsker stated and proved (not quite correctly) [4] a general extension for the Doob–Kolmogorov heuristic approach.
If is an -valued random variable whose probability distribution on is a tight measure then is said to be a separable random variable or a Radon random variable. Another equivalent criterion of the tightness of a collection M {\displaystyle M} is sequentially weakly compact.