Terminal σ-algebra

As a terminal σ-algebra or asymptotic σ-algebra or σ-algebra of terminal / asymptotic events , English tail σ-field , a special σ-algebra is referred to in probability theory . It is used in the investigation of limit values ​​and clearly contains all events, the occurrence of which does not change due to the modification of a finite number of sequence elements. The best-known application of the terminal σ-algebra is Kolmogorow's zero-one law .

definition

A measurement space and a sequence of sub-σ-algebras of . Then is called ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

${\displaystyle {\mathcal {T}}(({\mathcal {A}}_{n})_{n\in \mathbb {N} })=\bigcap _{k=1}^{\infty }\sigma \left(\bigcup _{l=k}^{\infty }{\mathcal {A}}_{l}\right)}$

the terminal σ-algebra of the sequence of σ-algebras or simply the terminal σ-algebra.

The terminal σ-algebra of a sequence of events is defined as the terminal σ-algebra of the sequence of σ-algebras . ${\displaystyle (A_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}_{n}:=\{A_{n},A_{n}^{C},\emptyset ,\Omega \}}$

The terminal σ-algebra of a sequence of random variables is defined as the terminal σ-algebra of the sequence of σ-algebras generated by the random variables. ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\sigma (X_{n}))_{n\in \mathbb {N} }}$

The notation for the terminal σ-algebra is not uniform in the literature. Sometimes it is with designated (for "asymptotically"), also found as well as a notation. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {T}}_{\infty },{\mathcal {G}}_{\infty },{\mathcal {E}}_{\infty }}$ ${\displaystyle \sigma _{\infty }}$

Constructive terms

Any set contained in terminal σ-algebra is called a terminal event or an asymptotic event .

A function that - - measure is called a terminal function . ${\displaystyle f\colon X\to {\overline {\mathbb {R} }}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {B}}({\overline {\mathbb {R} }})}$

Explanation

The meaning of the terminal σ-algebra becomes clearer by breaking the definition: the σ-algebra

${\displaystyle {\mathcal {C}}_{k}=\sigma \left(\bigcup _{l=k}^{\infty }{\mathcal {A}}_{l}\right)}$

contains by definition all sets that are contained in the σ-algebras for . ${\displaystyle {\mathcal {A}}_{l}}$ ${\displaystyle l\geq k}$

The terminal σ-algebra is now the intersection of all these set systems

${\displaystyle {\mathcal {T}}=\bigcap _{k=1}^{\infty }{\mathcal {C}}_{k}}$

and accordingly contains those quantities which are contained in all . Thus the terminal σ-algebra contains those events that do not depend on the first σ-algebras. Changing a finite number of σ-algebras does not change the terminal σ-algebra. ${\displaystyle {\mathcal {C}}_{k}}$ ${\displaystyle k}$

properties

• The terminal σ-algebra is not trivial in the sense that it contains more sets than just the superset and the empty set. For example, the limes superior and limes inferior of sequence of sets are terminal events, i.e. they are contained in the terminal σ-algebra. There are also nontrivial terminal functions. These include, for example, the superior and inferior limes of a sequence of random variables, as well as the limit values ​​of the Cesàro mean of random variables. ${\displaystyle \Omega }$
• One of the most important statements about the terminal σ-algebra is Kolmogorov's zero-one law . It says that if there are stochastically independent σ-algebras on the probability space , the terminal σ-algebra is a P-trivial σ-algebra , i.e. either or holds for every terminal event . ${\displaystyle ({\mathcal {A}}_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\Omega ,{\mathcal {A}},P)}$ ${\displaystyle A\in {\mathcal {T}}}$ ${\displaystyle P(A)=0}$ ${\displaystyle P(A)=1}$
• In addition, the terminal σ-algebra is always included in the interchangeable σ-algebra . If there is an interchangeable family of random variables , there is also a terminal event for each interchangeable event , so that . ${\displaystyle {\mathcal {E}}}$ ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle A\in {\mathcal {E}}}$ ${\displaystyle B\in {\mathcal {T}}}$ ${\displaystyle P(A\,\triangle \,B)=0}$

More general definitions

The above definition of the terminal σ-algebra is generalized in the literature as follows:

• It is not defined for sequences of σ-algebras, but more generally for sequences of arbitrary set systems . The terminal σ-algebra is then still a σ-algebra, but some statements remain incorrect without additional assumptions about the set systems. These statements also include Kolmogorov's zero-one law. ${\displaystyle {\mathcal {E}}_{n}\subset {\mathcal {A}}}$
• It is defined for any countably infinite index sets . The idea of ​​the above definition that terminal events are not influenced by the first k events is adapted in such a way that terminal events are not influenced by the modification of a finite number of events. Accordingly, the terminal σ-algebra is then defined as ${\displaystyle I}$

${\displaystyle {\mathcal {T}}(({\mathcal {A}}_{i})_{i\in I}):=\bigcap _{J\subset I \atop |J|<\infty }\sigma \left(\bigcup _{j\in I\setminus J}{\mathcal {A}}_{j}\right)}$.

literature

• 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 .
• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• 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 .

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 85.
2. ↑ Kusolitsch: Measure and probability theory. 2014, p. 51.
3. ^ Schmidt: Measure and probability. 2011, p. 234.
4. ↑ Klenke: Probability Theory. 2013, p. 64.