Weak convergence of measures

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It has been suggested that this article or section be merged into Convergence of measures. (Discuss)

In mathematics and statistics, weak convergence (also known as narrow convergence) is one of many types of convergence relating to the convergence of measures.

1 Equivalent definitions: the portmanteau theorem
2 Definition and notation
3 Weak convergence of random variables
4 References
5 See also

[edit] Equivalent definitions: the portmanteau theorem

There are (at least) five equivalent definitions of weak convergence of a sequence of measures, some of which are more general than others. The following equivalence result is sometimes known as the portmanteau theorem.

Let $(Ω, T)$ be a topological space with its Borel sigma algebra $σ(T)$ , and let $\mathcal{P}(\Omega)$ denote the collection of all probability measures defined on $(Ω,σ(T))$ . Let $(\mu_{n})_{n = 1}^{\infty}$ be a sequence in $\mathcal{P}(\Omega)$ and let $\mu \in \mathcal{P}(\Omega)$ . Then the following conditions are all equivalent:

$\lim_{n \to \infty} \int_{\Omega} f \, \mathrm{d} \mu_{n} = \int_{\Omega} f \, \mathrm{d} \mu$ for all bounded and continuous functions $f : \Omega \to \mathbb{R}$ ;
$\limsup_{n \to \infty} \mu_{n} (C) \leq \mu (C)$ for all closed sets $C \subseteq \Omega$ ;
$\liminf_{n \to \infty} \mu_{n} (U) \geq \mu (U)$ for all open sets $U \subseteq \Omega$ ;
$\lim_{n \to \infty} \mu_{n} (A) = \mu (A)$ for all " $μ$ -continuity" sets $A$ : sets $A \subseteq \Omega$ with $\mu (\partial A) = 0$ , where $\partial A$ denotes the boundary of $A$ ;
in the case $\Omega = \mathbb{R}$ , if $F n, F$ denote the cumulative distribution functions of the measures $μ n,μ$ respectively, then $\lim_{n \to \infty} F_{n} (x) = F(x)$ for all points $x \in \mathbb{R}$ at which $F$ is continuous.

[edit] Definition and notation

If any (and hence all) of the above conditions hold, the sequence of measures $(\mu_{n})_{n = 1}^{\infty}$ is said to converge weakly to $μ$ . Weak convergence is also known as narrow convergence, convergence in distribution and convergence in law (the terms "convergence in distribution/law" are more frequently used when discussing weak convergence of random variables, as in the next section).

There are many "arrow notations" for this kind of convergence: the most frequently used are $\mu_{n} \Rightarrow \mu$ , $\mu_{n} \rightharpoonup \mu$ and $\mu_{n} \, \begin{matrix} {\,}_\mathcal{D} \\ {\,}^{\longrightarrow} \\ \quad \end{matrix} \, \mu.$ .

[edit] Weak convergence of random variables

If $(\Omega, \mathcal{F}, \mathbb{P})$ is a probability space and $X_{n}, X : \Omega \to \mathbb{X}$ are random variables, $X n$ is said to converge weakly (or in distribution or in law) to $X$ as $n \to \infty$ if the sequence of push forward measures $(X_{n})_{*} (\mathbb{P})$ converges weakly to $X_{*} (\mathbb{P})$ in the sense of weak convergence of measures on $\mathbb{X}$ , as defined above.

[edit] References

Ambrosio, L., Gigli, N. & Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: ETH Zürich, Birkhäuser Verlag. ISBN 3-764-32428-7.
Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.
Billingsley, Patrick (1999). Convergence of Probability Measures. New York: John Wiley & Sons, Inc.. ISBN 0-471-19745-9.

Categories: Articles to be merged since November 2006 | Measure theory

Weak convergence of measures

From Wikipedia, the free encyclopedia

Contents

[edit] Equivalent definitions: the portmanteau theorem

[edit] Definition and notation

[edit] Weak convergence of random variables

[edit] References

[edit] See also

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