# Ionescu-Tulcea theorem

Ionescu-Tulcea theorem Not to be confused with: the Ionescu-Tulcea–Marinescu ergodic theorem.

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] Contents 1 Statement of the theorem 2 Applications 3 See also 4 Sources 5 References Statement of the theorem Suppose that {displaystyle (Omega _{0},{mathcal {A}}_{0},P_{0})} is a probability space and {displaystyle (Omega _{i},{mathcal {A}}_{i})} for {displaystyle iin mathbb {N} } is a sequence of measure spaces. For each {displaystyle iin mathbb {N} } let {displaystyle kappa _{i}colon (Omega ^{i-1},{mathcal {A}}^{i-1})to (Omega _{i},{mathcal {A}}_{i})} be the Markov kernel derived from {displaystyle (Omega ^{i-1},{mathcal {A}}^{i-1})} and {displaystyle (Omega _{i},{mathcal {A}}_{i}),} , where {displaystyle Omega ^{i}:=prod _{k=0}^{i}Omega _{k}{text{ and }}{mathcal {A}}^{i}:=bigotimes _{k=0}^{i}{mathcal {A}}_{k}.} Then there exists a sequence of probability measures {displaystyle P_{i}:=P_{0}otimes bigotimes _{k=1}^{i}kappa _{k}} defined on the product space for the sequence {displaystyle (Omega ^{i},{mathcal {A}}^{i})} , {displaystyle iin mathbb {N} ,} and there exists a uniquely defined probability measure {displaystyle P} on {displaystyle left(prod _{k=0}^{infty }Omega _{k},bigotimes _{k=0}^{infty }{mathcal {A}}_{k}right)} , so that {displaystyle P_{i}(A)=Pleft(Atimes prod _{k=i+1}^{infty }Omega _{k}right)} is satisfied for each {displaystyle Ain {mathcal {A}}^{i}} and {displaystyle iin mathbb {N} } . (The measure {displaystyle 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 Klenke, Achim (2013). Wahrscheinlichkeitstheorie (3rd ed.). Berlin Heidelberg: Springer-Verlag. pp. 292–294. doi:10.1007/978-3-642-36018-3. ISBN 978-3-642-36017-6. Kusolitsch, Norbert (2014). Maß- und Wahrscheinlichkeitstheorie: Eine Einführung (2nd ed.). Berlin; Heidelberg: Springer-Verlag. pp. 169–171. doi:10.1007/978-3-642-45387-8. ISBN 978-3-642-45386-1. References ^ Ionescu Tulcea, C. T. (1949). "Mesures dans les espaces produits". Atti Accad. Naz. Lincei Rend. 7: 208–211. ^ Shalizi, Cosma. "Chapter 3. Building Infinite Processes from Regular Conditional Probability Distributions" (PDF). Cosma Shalizi, CMU Statistics, Carnegie Mellon University. Index of /~cshalizi/754/notes "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". stat.cmu.edu/~cshalizi. ^ Jump up to: a b Abate, Alessandro; Redig, Frank; Tkachev, Ilya (2014). "On the effect of perturbation of conditional probabilities in total variation". Statistics & Probability Letters. 88: 1–8. arXiv:1311.3066. doi:10.1016/j.spl.2014.01.009. arXiv preprint Categories: Markov processesStochastic processes

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