Mixture (math)

The mixture of a measure-maintaining dynamic system is a term from ergodic theory , a branch of mathematics that is to be located between measure theory , the theory of dynamic systems and stochastics . One then speaks of mixing dimensionally maintaining dynamic systems , which are also called strongly mixing dimensionally maintaining dynamic systems , in order to distinguish them from a weakening of the term, weakly mixing dimensionally maintaining dynamic systems . In some cases, the mixture is also seen as a property of the dimensional preservation transformation , so one speaks of (strongly / weakly) mixing dimensional preserving images. Both strongly mixing and weakly mixing measure preserving systems are stronger terms than ergodic measure preserving dynamic systems and allow, for example, in the theory of stochastic processes a finer gradation of the range between independently identically distributed random variables and ergodic stochastic processes .

definition

A dimensionally preserving dynamic system with dimensionally preserving mapping is given . The dimensionally preserving dynamic system or the dimensionally preserving mapping is called (strong) mixing , if ${\displaystyle (\Omega ,{\mathcal {A}},\mu ,T)}$ ${\displaystyle T}$

${\displaystyle \lim _{n\to \infty }\mu (T^{-n}(A)\cap B)=\mu (A)\cdot \mu (B)}$

applies to all . The dimensionally preserving dynamic system or the dimensionally preserving mapping is called weakly mixing if ${\displaystyle A,B\in {\mathcal {A}}}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=1}^{n-1}\left|\mu (T^{-i}(A)\cap B)-\mu (A)\mu (B)\right|=0}$

applies to all . ${\displaystyle A,B\in {\mathcal {A}}}$

Relationship of mixture to ergodicity

The implications apply

${\displaystyle {\text{stark mischend}}\implies {\text{schwach mischend}}\implies {\text{ergodisch}}}$,

the inversions generally do not hold. The relationships are shown by means of the above definitions of the mixture and the following characterization of the ergodicity: is ergodic if and only if ${\displaystyle (\Omega ,{\mathcal {A}},\mu ,T)}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{i=0}^{n-1}\mu (T^{-i}(B)\cap A)=\mu (A)\cdot \mu (B)}$

is for everyone . ${\displaystyle A,B\in {\mathcal {A}}}$

Remarks

In stochastics, two sets are called stochastically independent if ${\displaystyle A,B}$

${\displaystyle P(A\cap B)=P(A)\cdot P(B)}$

applies. Thus the strong mixture can be understood as "asymptotic independence" from and for all sets of the σ-algebra. ${\displaystyle T^{-n}(A)}$ ${\displaystyle B}$

Web links

• DV Anosov: Mixing . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Manfred Einsiedler , Klaus Schmidt : Dynamic Systems . Ergodic theory and topological dynamics. Springer, Basel 2014, ISBN 978-3-0348-0633-6 , doi : 10.1007 / 978-3-0348-0634-3 .
• 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 .