Interchangeable family of random variables

“ Interchangeable family of random variables ” is a term from probability theory that formalizes the intuitive notion that the order of the evaluation does not matter when evaluating certain information. One of the most important statements about interchangeable families is de Finetti's representation theorem . Interchangeability is a weakening of the requirement that random variables be independently identically distributed .

definition

A family of random variables is called an interchangeable family of random variables if, for every permutation of the index set that only interchanges finitely many values ​​of , the distribution of coincides with the distribution of . ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle \sigma }$ ${\displaystyle I}$ ${\displaystyle I}$ ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle (X_{\sigma (i)})_{i\in I}}$

Equivalent to this is the definition that for all subsets with the distributions of are the same. ${\displaystyle J\subset I}$ ${\displaystyle |J|=n}$ ${\displaystyle (X_{i})_{i\in J}}$

Alternatively and equivalently, a family of random variables is defined as interchangeable if and only if for each and every pair of different elements exist, so that and are identically distributed. ${\displaystyle n}$ ${\displaystyle i_{1},\dots ,i_{n}\in I}$ ${\displaystyle j_{1},\dots ,j_{n}\in I}$ ${\displaystyle (X_{i_{1}},\dots ,X_{i_{n}})}$ ${\displaystyle (X_{j_{1}},\dots ,X_{j_{n}})}$

Comments and characteristics

• Interchangeable families are always distributed identically. This follows directly from the definition, since the equality of the distributions is required for all finite subsets and thus also for each individual random variable.
• A sequence of random variables is exchangeable if and only if a σ-algebra is given independently and identically . If this is the case, the terminal σ-algebra or the interchangeable σ-algebra can always be selected as the σ-algebra . This statement goes back to Bruno de Finetti . ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle {\mathcal {A}}}$

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 .
• Ulrich Krengel: Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , doi : 10.1007 / 978-3-663-09885-0 .