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.
A structure theorem for stochastic processes indexed by the discrete hypercube
