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-Lauchli Theorem.
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