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
Summary
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
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