Ionescu-Tulcea theorem explained

In the mathematical theory of probability, the Ionescu-Tulcea theorem, sometimes called the Ionesco Tulcea extension theorem, deals with the existence of probability measures for probabilistic events consisting of a countably infinite number of individual probabilistic events. In particular, the individual events may be independent or dependent with respect to each other. Thus, the statement goes beyond the mere existence of countable product measures. The theorem was proved by Cassius Ionescu-Tulcea in 1949.^{[1] [2]}

Statement of the theorem

Suppose that

(\Omega _{0,} l A _{0,} P _{0)} is a probability space and (\Omega _{i,} l A _{i)} for i \in \N is a sequence of measurable spaces. For each i \in \N let\kappa _{i} \colon (\Omega ^{i-1} , l A ^{i-1} ) \to (\Omega _{i,} l A _{i)} be the Markov kernel derived from (\Omega ^{i-1} , l A ^{i-1} ) and (\Omega _{i,} l A _{i),} , where\Omega ^{i\Omega} _{k} and l A ^{i:=} o times ^{i} _{k =0} l A _{k.}

Then there exists a sequence of probability measures

P _{i:=P} _{0} ⊗ o times ^{i} _{k =1} \kappa _{k} defined on the product space for the sequence (\Omega ^{i,} l A ^{i)} , i \in \N, and there exists a uniquely defined probability measure P on \left(\prod ^{inf ty} _{k =0} \Omega _{k,} o times ^{inf ty} _{k =0} l A _{k\right)} , so thatP _{i(A)=P\left(} A x \prod ^{inf ty} _{k = i +1} \Omega _{k} \right)

is satisfied for each

A \in l A ^{i} and i \in\N . (The measure P has conditional probabilities equal to the stochastic kernels.)^[3]

Applications

The construction used in the proof of the Ionescu-Tulcea theorem is often used in the theory of Markov decision processes, and, in particular, the theory of Markov chains.^[3]

See also

• Disintegration theorem
• Regular conditional probability

Sources

• Book: Klenke, Achim . Wahrscheinlichkeitstheorie . 3rd . Springer-Verlag . Berlin Heidelberg . 2013 . 978-3-642-36017-6 . 292–294 . 10.1007/978-3-642-36018-3.
• Book: Kusolitsch, Norbert . Maß- und Wahrscheinlichkeitstheorie: Eine Einführung . 2nd . Springer-Verlag . Berlin; Heidelberg . 2014 . 978-3-642-45386-1 . 169-171 . 10.1007/978-3-642-45387-8.

References

1. Ionescu Tulcea, C. T.. Mesures dans les espaces produits. Atti Accad. Naz. Lincei Rend.. 7. 1949. 208–211.
2. Web site: Shalizi, Cosma. Cosma Shalizi. Cosma Shalizi, CMU Statistics, Carnegie Mellon University. Chapter 3. Building Infinite Processes from Regular Conditional Probability Distributions. Index of /~cshalizi/754/notes Web site: stat.cmu.edu/~cshalizi. Almost None of the Theory of Stochastic Processes: A Course on Random Processes, for Students of Measure-Theoretic Probability, with a View to Applications in Dynamics and Statistics by Cosma Rohilla Shalizi with Aryeh Kontorovich.
3. Abate, Alessandro. Redig, Frank. Tkachev, Ilya. On the effect of perturbation of conditional probabilities in total variation. 2014. Statistics & Probability Letters. 88. 1–8. 10.1016/j.spl.2014.01.009. 1311.3066. arXiv preprint