Measurable events indexed by trees
Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros

TL;DR
This paper investigates the behavior of measurable events indexed by homogeneous trees in probability spaces, establishing conditions under which intersections of events have a lower bounded probability, with explicit bounds depending on the tree's branching number.
Contribution
It introduces a new combinatorial result on measurable events indexed by homogeneous trees, providing explicit bounds and a finite version of the theorem.
Findings
Existence of strong subtrees with controlled intersection probabilities
Explicit formula for the bound q(b,n)
Finite version of the main result
Abstract
A tree is said to be homogeneous if it is uniquely rooted and there exists an integer , called the branching number of , such that every has exactly immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer and every integer there exists an integer with the following property. If is a homogeneous tree with branching number and is a family of measurable events in a probability space satisfying for every , then for every there exists a strong subtree of of infinite height such that for every non-empty finite subset of of cardinality we have \[ \mu\Big(\bigcap_{t\in F} A_t\Big) \meg \theta^{q(b,n)}. \] In fact, we…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
