Ionescu-Tulcea theorem

The Ionescu-Tulcea theorem is a mathematical theorem of probability theory that deals with the existence of probability measures for probability experiments that consist of countably infinite individual experiments. In particular, the individual experiments can be different and dependent on one another . Thus the statement goes beyond the mere existence of countable product dimensions . The theorem was proved by Cassius Ionescu-Tulcea .

statement

A probability space and measurement spaces for . With the notation ${\displaystyle (\Omega _{0},{\mathcal {A}}_{0},P_{0})}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in \mathbb {N} }$

${\displaystyle \Omega ^{i}:=\prod _{k=0}^{i}\Omega _{k}{\text{ und }}{\mathcal {A}}^{i}:=\bigotimes _{k=0}^{i}{\mathcal {A}}_{k}}$

are Markov kernels of according given for . Then they exist through the product of the kernels ${\displaystyle \kappa _{i}}$ ${\displaystyle (\Omega ^{i-1},{\mathcal {A}}^{i-1})}$ ${\displaystyle (\Omega _{i},{\mathcal {A}}_{i})}$ ${\displaystyle i\in \mathbb {N} }$

${\displaystyle P_{i}:=P_{0}\otimes \bigotimes _{k=0}^{i}\kappa _{k}}$

defined probability measures on and there is a clearly defined probability measure on , so ${\displaystyle (\Omega ^{i},{\mathcal {A}}^{i})}$ ${\displaystyle P}$ ${\displaystyle (\prod _{k=0}^{\infty }\Omega _{k},\bigotimes _{k=0}^{\infty }{\mathcal {A}}_{k})}$

${\displaystyle P_{i}(A)=P\left(A\times \prod _{k=i+1}^{\infty }\Omega _{k}\right)}$

applies to everyone and . ${\displaystyle A\in {\mathcal {A}}^{i}}$ ${\displaystyle i\in \mathbb {N} }$

use

The Ionescu-Tulcea theorem is widely used. For example, it provides the existence of any discrete time stochastic processes . Alternatively, it can also be used to show the existence of infinite product measures or of countable families of stochastically independent random variables .

Generalizations

A generalization of the Ionescu-Tulcea theorem is Kolmogorow's extension theorem , which deals with the existence of probability measures on uncountable product spaces.

literature

• Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 292-294 , 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 , p. 169-171 , doi : 10.1007 / 978-3-642-45387-8 .