A structure theorem for stochastic processes indexed by the discrete hypercube

Pandelis Dodos; Konstantinos Tyros

paper

A structure theorem for stochastic processes indexed by the discrete hypercube

arXiv:1808.10352

Abstract

Let $A$ be a finite set with $∣ A ∣ ⩾ 2$ , let $n$ be a positive integer, and let $A^{n}$ denote the discrete $n$ -dimensional hypercube (that is, $A^{n}$ is the Cartesian product of $n$ many copies of $A$ ). Given a family $⟨ D_{t} : t \in A^{n} ⟩$ of measurable events in a probability space (a stochastic process), what structural information can be obtained assuming that the events $⟨ D_{t} : t \in A^{n} ⟩$ 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.