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Measurable Events Indexed by Trees

Published online by Cambridge University Press:  12 March 2012

PANDELIS DODOS ,
VASSILIS KANELLOPOULOS  and
KONSTANTINOS TYROS
PANDELIS DODOS
Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis 157 84, Athens, Greece (e-mail: pdodos@math.uoa.gr)
VASSILIS KANELLOPOULOS
Affiliation:
National Technical University of Athens, Faculty of Applied Sciences, Department of Mathematics, Zografou Campus, 157 80, Athens, Greece (e-mail: bkanel@math.ntua.gr)
KONSTANTINOS TYROS
Affiliation:
Department of Mathematics, University of Toronto, Toronto, CanadaM5S 2E4 (e-mail: k.tyros@utoronto.ca)
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Abstract

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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. We study the behaviour of measurable events in probability spaces indexed by homogeneous trees.

Precisely, we show that for every integer b ≥ 2 and every integer n ≥ 1 there exists an integer q(b,n) with the following property. If T is a homogeneous tree with branching number b and {At:t ∈ T} is a family of measurable events in a probability space (Ω,Σ,μ) satisfying μ(At)≥ϵ>0 for every t ∈ T, then for every 0<θ<ϵ there exists a strong subtree S of T of infinite height, such that for every finite subset F of S of cardinality n ≥ 1 we have \[ \mu\biggl( \bigcap_{t\in F} A_t\biggr) \meg \theta^{q(b,n)}. \] In fact, we can take q(b,n)= ((2b−1)2n−1−1)·(2b−2)−1. A finite version of this result is also obtained.

Keywords

Primary 05D1005C05

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 3 , May 2012 , pp. 374 - 411
Copyright
Copyright © Cambridge University Press 2012

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