Distribution function (measure theory)

The distribution function of a measure is a term from measure theory , a branch of mathematics that deals with generalized length and volume concepts. A distribution function can be assigned to every finite measure on the real numbers . Distribution functions of probability measures play an important role in stochastics . In measure theory, distribution functions are used to check the convergence of measures.

definition

Given the measuring room , with the Borel σ-algebra called, and a finite measure on this measuring space. Then is called ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle \mu }$

${\displaystyle F_{\mu }(x):=\mu ((-\infty ,x])}$

the distribution function of the measure . ${\displaystyle \mu }$

In addition, every monotonically growing , right-hand continuous and bounded real function is called a distribution function, since it is through ${\displaystyle F}$

${\displaystyle \mu _{F}((a,b]):=F(b)-F(a)}$

defines a finite measure. A special case are those functions for which additionally applies

${\displaystyle \lim _{x\to -\infty }F(x)=0{\text{ und }}\lim _{x\to \infty }F(x)=1}$,

these are precisely the distribution functions in the sense of probability theory .

Examples

Looking at the Dirac dimension on the 1st

${\displaystyle \delta _{1}(A):={\begin{cases}1&{\text{falls }}1\in A\\0&{\text{falls }}1\notin A\end{cases}}}$

Then the distribution function is

${\displaystyle F_{\delta _{1}}={\begin{cases}0&{\text{falls }}x<1\\1&{\text{falls }}x\geq 1\end{cases}}}$.

properties

• If one defines an equivalence relation on the monotonically growing, right-sided continuous and bounded functions

${\displaystyle F\sim G\iff F-G{\text{ ist konstant }}}$

and denotes the equivalence classes with , then it is a bijection. Each finite measure on the real numbers is assigned the equivalence class of its distribution function. Therefore, one usually does not differentiate between the measure and the distribution function. This equivalence class formation is not necessary for distribution functions in the sense of probability theory, since they are already clearly and clearly defined. ${\displaystyle [F]}$ ${\displaystyle \mu \mapsto [F_{\mu }]}$ ${\displaystyle \lim _{x\to -\infty }F(x)=0}$ ${\displaystyle \lim _{x\to \infty }F(x)=1}$

• If you set

${\displaystyle \Vert F_{\mu }\Vert ^{*}:=\lim _{x\to \infty }\left(F_{\mu }(x)-F_{\mu }(-x)\right)}$,

so is . The total variation norm denotes ${\displaystyle \Vert F_{\mu }\Vert ^{*}=\Vert \mu \Vert _{TV}}$ ${\displaystyle \Vert \cdot \Vert _{TV}}$

convergence

Vague convergence

A sequence of distribution functions is called vaguely convergent to the distribution function if it converges from pointwise to at all points of continuity, i.e. if ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle F}$

${\displaystyle \lim _{n\to \infty }F_{n}(x)=F(x)}$

applies to all where is continuous. ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle F}$

Weak convergence

→ Main article : Weak convergence (measure theory)

A sequence of distribution functions is said to be weakly convergent to the distribution function if it is vaguely convergent and ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$

${\displaystyle \lim _{n\to \infty }\Vert F_{n}\Vert ^{*}=\Vert F\Vert ^{*}}$

applies.

If the distribution functions belong to probability measures, then the second condition can be dispensed with, since then always applies. Thus, weak and vague convergence then coincide. For probability measures, the weak convergence of the distribution functions can be metrized with the Lévy distance . ${\displaystyle \|F_{n}\|^{*}=1}$

comment

The weak and vague convergence of distribution functions is not clearly used in the literature. Sometimes there is no differentiation between vague and weak convergence, since these terms for probability measures coincide, sometimes the point-by-point convergence at all points of continuity is also referred to as weak convergence. This would correspond to the vague convergence described here. For distribution functions in the sense of probability theory , which are defined via real random variables , there is also the term convergent in distribution or stochastically convergent .

Important sentences

Helly-Bray's theorem

According to Helly-Bray's theorem:

• Converges a sequence of distribution functions vaguely against so converges vague in the sense of measure theory against . ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$
• Converges a sequence of distribution functions weak against , converges slightly in the sense of measure theory against . ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle \mu _{n}}$ ${\displaystyle \mu }$

Modifying the series of distribution functions with a series of real numbers also shows the reverse direction.

Selection set from Helly

According to Helly's selection theorem , every uniformly bounded sequence of distribution functions has a vaguely convergent subsequence.

Prokhorov theorem

The set of Prokhorov can be specially formulated for (even limited) families of distribution functions. It says that a family of distribution functions is tight if and only if every sequence in this family has a weakly convergent subsequence.

Individual evidence

1. ↑ Kusolitsch: Measure and probability theory. 2014, p. 287.

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 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer-Verlag, Heidelberg / Dordrecht / London / New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .