Distances Between σ-Fields on a Probability Space
Andrzej Komisarski ()
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Andrzej Komisarski: University of Łódź
Journal of Theoretical Probability, 2008, vol. 21, issue 4, 812-823
Abstract:
Abstract For a probability space (Ω,ℱ,P) and two sub-σ-fields $\mathcal{A},\mathcal{B}\subset\mathcal{F}$ we consider two natural distances: $\rho(\mathcal{B},\mathcal{A})=\sup_{B\in\mathcal{B}}\inf_{A\in\mathcal{A}}P(A\triangle B)$ and $\overline{\rho}(\mathcal{B},\mathcal{A})=\sup_{B\in\mathcal{B}}\inf_{A\in\mathcal{A},A\supset B}P(A\setminus B)$ . We investigate basic properties of these distances. In particular we show that if a distance (ρ or $\overline{\rho}$ ) from ℬ to $\mathcal{A}$ is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists $A\in\mathcal{A}$ such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).
Keywords: Hausdorff metric of σ-fields; Distance between σ-algebras; 60A10 (search for similar items in EconPapers)
Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:spr:jotpro:v:21:y:2008:i:4:d:10.1007_s10959-008-0149-7
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DOI: 10.1007/s10959-008-0149-7
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