Abstract

Abstract Let A be a finite set with , let n be a positive integer, and let $A^n$ denote the discrete $n\text {-dimensional}$ hypercube (that is, $A^n$ is the Cartesian product of n many copies of A). Given a family $\langle D_t:t\in A^n\rangle $ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $\langle D_t:t\in A^n\rangle $ are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to a mild ‘stationarity’ condition. Our result has a number of combinatorial consequences, including a new (and the most informative so far) proof of the density Hales-Jewett theorem.

Highlights

  • We wish to compare the distributions of the random variables

  • (2) We say that a stochastic process : ∈ in a probability space (Ω, F, P) is (, )-insensitive provided that = for every, ∈ that are (, )-equivalent

  • Its most important feature is the quantitative improvement of a crucial step that appears in all known combinatorial proofs of the density Hales-Jewett theorem. (We discuss this particular feature in Remark 6.) The driving force behind this improvement is Theorem 2

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Summary

Introduction

Let be a nonempty finite set and let : ∈ and : ∈ be stochastic processes (families of measurable events) in a probability space (Ω, F, P) with P( ) = P( ) = > 0 for all ∈ , and assume that the events : ∈ are independent. A classical method for doing so is by comparing the moments of X and Y (see, e.g., [Du]), a task that essentially reduces to that of comparing the joint probability of : ∈ with the expected value | | as varies over all nonempty subsets of the index set. Assuming that the random variables X and Y are not close in distribution, one is led to the following problem This is because X and Y are both sums of indicator functions

Pandelis Dodos and Konstantinos Tyros
Deviating from the expected value
The main result
Correlations over arbitrary sets
Outline of the argument
Combinatorial background
Colorings of combinatorial lines
Correlations over combinatorial lines
Stationarity
Insensitivity
Proof of the density Hales-Jewett theorem
Step 1
Step 2
Step 3
Comments
The type of a subset of a discrete hypercube
The type of a nonempty tuple
The type of a nonempty finite set
Types and the Ramsey property
Stochastic processes and types
The separation index
Correlations over 1-separated sets
Proof of Proposition 3
Proof of Theorem 5
Obstructions to independence
Proof of Proposition 4
Full Text
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