Measures of correlation and mixtures of product measures

Total correlation (`TC') and dual total correlation (`DTC') are two classical measures of correlation for an $n$-tuple of random variables. They both reduce to mutual information when $n=2$. The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution $\mu$ has small TC or DTC. If $\mathrm{TC}(\mu) = o(n)$, then $\mu$ is close in the transportation metric to a product measure: this follows quickly from Marton's classical transportation-entropy inequality. On the other hand, if $\mathrm{DTC}(\mu) = o(n)$, then the structural consequence is more complicated: $\mu$ is close to a mixture of a controlled number of terms, most of them close to product measures in the transportation metric. This is the main new result of the paper.