A Structure Theorem For Stochastic Processes Indexed By The Discrete Hypercube

Let A be a finite set with vertical bar A vertical bar >= 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 is an element of 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 is an element of A(n)> are not behaving as if they were independent? We obtain an answer to this problem (in a strong quantitative sense) subject to amild '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.