Interchangeable σ-algebra

The exchangeable σ-algebra is a special set system in stochastics , the elements of which are invariant under certain permutations . Interchangeable σ-algebras occur, for example, in the context of interchangeable families of random variables or the 0-1 law of Hewitt-Savage .

definition

A stochastic process is given , where each has values ​​in . Be ${\displaystyle X=(X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X_{n}}$ ${\displaystyle E}$

${\displaystyle \mathbf {F} _{n}:=\{f\,|\,f:E^{\mathbb {N} }\to \mathbb {R} {\text{ ist messbar und n-symmetrisch }}\}}$

the set of all measurable n-symmetric maps .

Define

${\displaystyle {\mathcal {S}}_{n}:=\sigma (\mathbf {F} _{n})}$

the σ-algebra generated by these functions. Then

${\displaystyle {\mathcal {E}}_{n}:=X^{-1}({\mathcal {S}}_{n})}$

the σ-algebra of all events invariant under permutations of the first indices of the stochastic process. The interchangeable σ-algebra is then defined as ${\displaystyle n}$

${\displaystyle {\mathcal {E}}=\bigcap _{n=1}^{\infty }{\mathcal {E}}_{n}}$

and thus the σ-algebra of all events invariant under permutations of finitely many indices of the stochastic process.

Relationship to the terminal σ-algebra

The terminal σ-algebra is always contained in the interchangeable σ-algebra, because with the representation for the terminal σ-algebra

${\displaystyle {\mathcal {T}}=\bigcap _{n\in \mathbb {N} }\sigma (X_{n+1},X_{n+2},\dots )}$

is always

${\displaystyle \sigma (X_{n+1},X_{n+2},\dots )\subset {\mathcal {E}}_{n}}$

and thus

${\displaystyle {\mathcal {T}}=\bigcap _{n\in \mathbb {N} }\sigma (X_{n+1},X_{n+2},\dots )\subset \bigcap _{n\in \mathbb {N} }{\mathcal {E}}_{n}={\mathcal {E}}}$.

Examples can also be constructed in which the interchangeable σ-algebra contains sets that are not contained in the terminal σ-algebra. The exchangeable σ-algebra is then genuinely larger than the terminal σ-algebra.

Conversely, it can be shown that for an interchangeable family of random variables there is a terminal event for every set , so that (the opposite conclusion is trivial because of ). For every set from the exchangeable σ-algebra, there is a set in the terminal σ-algebra, so that the difference becomes a zero set. ${\displaystyle X=(X_{1},X_{2},\dots )}$ ${\displaystyle A\in {\mathcal {E}}}$ ${\displaystyle B\in {\mathcal {T}}}$ ${\displaystyle P(A\,\triangle \,B)=0}$ ${\displaystyle {\mathcal {T}}\subset {\mathcal {E}}}$

From this one can immediately deduce Hewitt-Savage's 0-1 law , namely that the exchangeable σ-algebra of an independently identically distributed sequence of random variables is a P-trivial σ-algebra . According to Kolmogorov's zero-one law , the terminal σ-algebra is P-trivial and, based on the above result, also the exchangeable σ-algebra.

literature

• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 237-247 , doi : 10.1007 / 978-3-642-36018-3 .