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## Uniformity for the decay of correlations

Decay of correlations at a speed given by some numbers $u_n\to0$ for some dynamics $T:X\to X$ and two spaces $E,F$ of functions over $X$ with zero average, is formulated in two similar ways in various works: for all $f\in E,\; g\in F$,

1. $|cov(f,g\circ T^n)| \leq C(f,g) u_n$ for some $C(f,g)<\infty$ depending arbitrarily on $f,g$;
2. $|cov(f,g\circ T^n)| \leq K \|f\|\cdot \|g\| u_n$ for some $K<\infty$ independent of $f,g,n$;

where $cov(f,g\circ T^n)$ is the correlation at time $n\geq0$, a bilinear form $E\times F\to\mathbb R$.

The second type seems stronger than the first (and it is strictly so for arbitrary families of bilinear forms). However the above two types are really equivalent under very general assumptions, i.e., if (i) $E,F$ are Banach spaces and (ii) each $(f,g)\mapsto cov(f,g\circ T^n)$ is continuous for any given $n\geq0$.

Indeed, fix $f\in E$.Let $c_{f,n}:F\to\mathbb R$ be defined by $c_{f,n}(g)=cov(f,g\circ T^n)/u_n$. Each such map is linear and continuous by (ii). Now, for each $g\in F$, $\sup_n \| c_{f,n}(g)\| \leq C(f,g)<\infty$. Using (i), the Banach-Steinhaus theorem gives $K_f<\infty$ such that $\sup_n \| c_{f,n}(g)\| \leq K_f \|g\|$. In other words, $c_f:F\to\ell^\infty(\mathbb N)$, defined by $c_f(g)=(c_{f,n}(g))_{n\geq0}$ is linear and continuous.

Similarly $c^g:E\to\ell^\infty(\mathbb N)$ defined by $c^g(f)=c_f(g)$ is linear and continuous. Now, for each $f\in E$, $\sup_{g\in F(1)} \|c^g(f)\| \leq K_f$ where $F(1)$ is the unit ball in $F$. Using (i), the Banach-Steinhaus theorem gives $K_*<\infty$ such that $\|c^g(f)\|\leq K_*\|f\|$ for all $g\in F(1)$. But this implies the first type of decay above. CQFD.

This was pointed out to me by Sébastien GOUËZEL.

Added:

Jean-René CHAZOTTES points out that this is essentially Theorem B.1 of his paper joint with P. Collet and B. Schmitt: Statistical consequences of the Devroye inequality for processes. Applications to a class of non-uniformly hyperbolic dynamical systems. Nonlinearity 18 (2005).

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