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

Rémi Peyre¹

¹ Institut Élie Cartan Université de Lorraine 54506 Vandœuvre-lès-Nancy, France

Studia Mathematica, Tome 216 (2013) no. 3, pp. 245-270

Résumé

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.

Détail

DOI : 10.4064/sm216-3-4

Keywords: sigma algebras mathcal mathcal rho mixing coefficient rho mathcal mathcal between mathcal mathcal supremum correlation between real random variables which mathcal resp mathcal measurable tau mathcal mathcal coefficient defined similarly restricting where indicator functions has known long time bound rho leq tau mathopen log tau holds constant article works best possible