Measurable events indexed by trees

Details Measurable events indexed by trees Measurable events indexed by trees

2011-05-12

arxiv.org/abs/1105.2417v1

Mathematics

Combinatorics

38 pages, 2 figures

Scientific paper

A tree $T$ is said to be homogeneous if it is uniquely rooted and there exists an integer $b\geq 2$, called the branching number of $T$, such that every $t\in T$ has exactly $b$ immediate successors. We study the behavior of measurable events in probability spaces indexed by homogeneous trees. Precisely, we show that for every integer $b\geq 2$ and every integer $n\geq 1$ there exists an integer $q(b,n)$ with the following property. If $T$ is a homogeneous tree with branching number $b$ and $\{A_t:t\in T\}$ is a family of measurable events in a probability space $(\Omega,\Sigma,\mu)$ satisfying $\mu(A_t)\geq\epsilon>0$ for every $t\in T$, then for every $0<\theta<\epsilon$ there exists a strong subtree $S$ of $T$ of infinite height such that for every non-empty finite subset $F$ of $S$ of cardinality $n$ we have \[ \mu\Big(\bigcap_{t\in F} A_t\Big) \meg \theta^{q(b,n)}. \] In fact, we can take $q(b,n)= \big((2^b-1)^{2n-1}-1\big)\cdot(2^b-2)^{-1}$. A finite version of this result is also obtained.

Affiliated with

Also associated with

No associations

Rating

Measurable events indexed by trees does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Measurable events indexed by trees, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Measurable events indexed by trees will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-496798