  Let $A$ be a finite set with $|A|\geqslant 2$, 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 $\langle
D_t:t\in A^n\rangle$ 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.
