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Abstract

For two $\sigma $-algebras $\mathcal {A}$ and $\mathcal {B}$, the $\rho $-mixing coefficient $\rho (\mathcal {A},\mathcal {B})$ between $\mathcal {A}$ and $\mathcal {B}$ is the supremum correlation between two real random variables $X$ and $Y$ which are $\mathcal {A}$- resp. $\mathcal {B}$-measurable; the $\tau '(\mathcal {A},\mathcal {B})$ coefficient is defined similarly, but restricting to the case where $X$ and $Y$ are indicator functions. It has been known for a long time that the bound $\rho \leq C\tau '(1+\mathopen |\log\tau '|)$ holds for some constant $C$; in this article, we show that $C = 1$ works and is best possible.