Geodesic
A structure theorem for stochastic processes indexed by the discrete hypercube
Forum of Mathematics, Sigma, Tome 9 (2021)

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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. 
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     author = {Pandelis Dodos and Konstantinos Tyros},
     title = {A structure theorem for stochastic processes indexed by the discrete hypercube},
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Pandelis Dodos; Konstantinos Tyros. A structure theorem for stochastic processes indexed by the discrete hypercube. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.3

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