Multi-variate correlation and mixtures of product measures

Total correlation (`TC') and dual total correlation (`DTC') are two classical ways to quantify the correlation among 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 $μ$ has small TC or DTC. If $\mathrm{TC}(μ) = o(n)$, then $μ$ is close to a product measure according to a suitable transportation metric: this follows directly from Marton's classical transportation-entropy inequality. If $\mathrm{DTC}(μ) = o(n)$, then the structural consequence is more complicated: $μ$ is 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.

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