Sharp equivalence between $\rho $- and $\tau $-mixing coefficients

Tom 216 / 2013

Rémi Peyre Studia Mathematica 216 (2013), 245-270 MSC: Primary 62H20. DOI: 10.4064/sm216-3-4


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.


  • Rémi PeyreInstitut Élie Cartan
    Université de Lorraine
    54506 Vandœuvre-lès-Nancy, France

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