Weak Convergence (Measure Theory)

The weak convergence is a concept of measure theory , a branch of mathematics that deals with generalized length and volume terms. Weak convergence is a concept of convergence for finite measures and contains the convergence in distribution of probability theory as a special case . A modification for measures on function spaces is the fdd convergence .

definition

Let be a metric space and the Borel σ-algebra as well as the set of finite measures on the measurement space . Be off . Is ${\displaystyle (X,d)}$ ${\displaystyle {\mathcal {B}}(X)}$ ${\displaystyle {\mathcal {M}}_{f}^{+}(X)}$ ${\displaystyle (X,{\mathcal {B}}(X))}$ ${\displaystyle \mu ,\mu _{n}}$ ${\displaystyle {\mathcal {M}}_{f}^{+}(X)}$

${\displaystyle \lim _{n\to \infty }\int _{X}f\mathrm {d} \mu _{n}=\int _{X}f\mathrm {d} \mu }$

for all bounded continuous functions , then is called weakly convergent to . It then writes also weak, or . The "w" stands for "weakly". ${\displaystyle f}$ ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$ ${\displaystyle \mu _{n}\rightarrow \mu }$ ${\displaystyle \mu _{n}{\xrightarrow[{}]{w}}\mu }$ ${\displaystyle \mu =w{\text{-}}\!\lim _{n\to \infty }\mu _{n}}$

Motivation to define

Intuitively one would say of a sequence of measures that it converges to if ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \mu }$

${\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}$

holds for every set from the considered σ-algebra. But if you now use the measuring room as a sequence of dimensions, for example ${\displaystyle A}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle \mu _{n}(A)=\delta _{\tfrac {1}{n}}(A)}$

the Dirac measures in each case in the point , one would "intuitively" expect that the sequence converges to the Dirac measure in the point . But this is not the case, as you can see from the crowd , for example , because it is ${\displaystyle {\tfrac {1}{n}}}$ ${\displaystyle \delta _{0}}$ ${\displaystyle 0}$ ${\displaystyle A=(-\infty ,0]}$

${\displaystyle \lim _{n\to \infty }\delta _{\tfrac {1}{n}}(A)=0\neq \delta _{0}(A)=1}$.

So the concept of convergence is too strong. An equivalent formulation of the above, intuitive concept of convergence for sequences of measures is

${\displaystyle \lim _{n\to \infty }\int _{X}f\mathrm {d} \mu _{n}=\int _{X}f\mathrm {d} \mu }$

for all , i.e. the essentially limited functions . On the basis of this characterization, one now looks for weaker functional classes and sets of measures , so that the above equation still applies to this choice and is also a separating family for . So it should also ${\displaystyle f\in {\mathcal {L}}^{\infty }(X,{\mathcal {B}}(X),\mu )}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \int _{X}f\mathrm {d} \mu =\int _{X}f\mathrm {d} \nu {\text{ für alle }}f\in {\mathcal {F}}\implies \mu =\nu }$

be valid. This guarantees the uniqueness of the limit value. If one now chooses the finite measures and the bounded continuous functions, one obtains the weak convergence described here. Another choice of the function classes and sets of measures provides, for example, the vague convergence or the convergence in distribution of the stochastics. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {F}}}$

Relationship to other types of convergence

Relationship to convergence with respect to the norm of total variation

If one considers the set of finite measures as a subset of the vector space of the finite signed measures provided with the total variation norm as the norm, then the convergence with respect to the total variation norm and the weak convergence can be related. The weak convergence then always follows from the convergence with respect to the total variation norm, because it is

${\displaystyle \left|\int f\mathrm {d} \mu _{n}-\int f\mathrm {d} \mu \right|\leq \int |f|\mathrm {d} |\mu _{n}-\mu |\leq \|f\|_{\infty }\|\mu _{n}-\mu \|_{TV}}$

for all bounded continuous functions. Here the variation and the total variation norm of the measure . ${\displaystyle |\mu |}$ ${\displaystyle \|\mu \|_{TV}}$ ${\displaystyle \mu }$

Customized relationship to convergence

The convergence to measure and the weak convergence can be about the convergence of image dimensions link: Are measurable functions from a finite measure space into a separable metric space with the corresponding Borel σ-algebra, and converge to measure against , then converge on the measurement space the Image dimensions weak against . ${\displaystyle f_{n},f}$ ${\displaystyle (X,{\mathcal {A}},\mu )}$ ${\displaystyle (Y,d)}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle (Y,{\mathcal {B}}(Y))}$ ${\displaystyle f_{n}(\mu )}$ ${\displaystyle f(\mu )}$

Important sentences and statements

• The Portmanteau theorem enumerates various equivalent characterizations of the weak convergence of measures.
• According to Helly-Bray's theorem , a sequence of real finite measures converges to weakly if the distribution functions converge weakly . ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$
• In general, the weak convergence follows from the convergence with respect to the Prokhorov metric . If the base space is a separable space , the two types of convergence are equivalent.
• The weak convergence is given the measure of the basic set. This is done in the definition. Thus, weak limit values ​​of sequences of (sub) probability measures are again (sub) probability measures. ${\displaystyle f\equiv 1}$

classification

In functional analysis , weak convergence is understood to mean the following: starting from a standardized vector space (here the space of the signed dimensions , provided with the total variation norm ), the topological dual space is formed ${\displaystyle V}$

${\displaystyle V':=\{T\;|\;T\colon V\to \mathbb {K} {\text{ ist linear und stetig }}\}}$.

A sequence in is then called weakly convergent to if ${\displaystyle (x_{n})_{n\in N}}$ ${\displaystyle V}$ ${\displaystyle x\in V}$

${\displaystyle \lim _{n\to \infty }T(x_{n})=T(x){\text{ für alle }}T\in V'}$

is. In a concrete case, this would be equivalent to being restricted and ${\displaystyle (\mu _{n})_{n\in \mathbb {N} }}$

${\displaystyle \lim _{n\to \infty }\mu _{n}(A)=\mu (A)}$

for all measurable . However, as has already been shown above, this is generally wrong, according to the Portmanteau theorem it only applies to borderless sets . Thus the weak convergence described here is really weaker than the weak convergence in the sense of functional analysis. ${\displaystyle A}$

In fact, the concept of weak convergence of measures is much more like weak - * convergence than weak convergence. One starts again from a normalized vector space (this time the space of the continuously limited functions, provided with the supremum norm ) and the topological dual space . A sequence from the dual space is then called weak - * - convergent to if ${\displaystyle V}$ ${\displaystyle V'}$ ${\displaystyle (T_{n})_{n\in \mathbb {N} }}$ ${\displaystyle T}$

${\displaystyle \lim _{n\to \infty }T_{n}(x)=T(x){\text{ für alle }}x\in V}$.

But since in this concrete case every finite measure for through ${\displaystyle f\in C_{b}(X)}$

${\displaystyle f\mapsto \int f\mathrm {d} \mu }$

forms a continuous linear form, the finite measures are in any case a subset of the dual space and the weak convergence of measures is a modification of the weak - * - convergence in the sense of functional analysis.

Weak topology

The topology generated by the weak convergence is called the weak topology , even if it corresponds more closely to the weak - * topology according to the above discussion . It is the coarsest topology, so that all mappings ${\displaystyle \tau _{S}}$

${\displaystyle {\mathcal {M}}_{f}^{+}(X)\to \mathbb {R} }$

defined by

${\displaystyle \mu \mapsto \int f\mathrm {d} \mu {\text{ für ein }}f\in C_{b}(X)}$

are steady. According to the above properties, is weaker than the topology generated by the total variation distance. It is also weaker on any metric spaces than the topology generated by the Prokhorov metric . If a separable metric space, then is equivalent to , since then the weak convergence and the convergence with respect to the Prokhorov metric are equivalent. Accordingly metrisiert the Prokhorov metric the weak convergence in this case. ${\displaystyle \tau _{S}}$ ${\displaystyle \tau _{P}}$ ${\displaystyle X}$ ${\displaystyle \tau _{S}}$ ${\displaystyle \tau _{P}}$

It is also Hausdorffsch, that is , a Hausdorff room . An environment base of is formed by ${\displaystyle ({\mathcal {M}}_{f}^{+}(X),\tau _{S})}$ ${\displaystyle \mu _{0}\in {\mathcal {M}}_{f}^{+}(X)}$

${\displaystyle U_{\epsilon ;f_{1},\dots ,f_{n}}(\mu _{0})=\{\mu \in {\mathcal {M}}_{f}^{+}(X)\,|\,\left|\int f_{j}\mathrm {d} \mu -\int f_{j}\mathrm {d} \mu _{0}\right|<\epsilon {\text{ für alle }}i=1,\dots ,n\}}$,

where they are. ${\displaystyle f_{i}\in C_{b}(X)}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .

Individual evidence

1. ↑ Klenke: Probability Theory. 2013, p. 257.