Measurable events indexed by products of trees |
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Authors: | Pandelis Dodos Vassilis Kanellopoulos Konstantinos Tyros |
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Institution: | 1. Department of Mathematics, University of Athens, Panepistimiopolis, 157 84, Athens, Greece 2. Faculty of Applied Sciences, Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece 3. Department of Mathematics, University of Toronto, Toronto, Canada, M5S 2E4
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Abstract: | A tree T is said to be homogeneous if it is uniquely rooted and there exists an integer b ≥ 2, called the branching number of T, such that every t ∈ T has exactly b immediate successors. A vector homogeneous tree T is a finite sequence (T 1,...,T d ) of homogeneous trees and its level product ?T is the subset of the Cartesian product T 1×...×T d consisting of all finite sequences (t 1,...,t d ) of nodes having common length. We study the behavior of measurable events in probability spaces indexed by the level product ?T of a vector homogeneous tree T. We show that, by refining the index set to the level product ?S of a vector strong subtree S of T, such families of events become highly correlated. An analogue of Lebesgue’s density Theorem is also established which can be considered as the “probabilistic” version of the density Halpern-Läuchli Theorem. |
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