Strong, uniform, and maximal-correlation mixing coefficients

Let $Y=(Y_k{)}_{k\ge 0}$ be a process on a probability space whose finite-dimensional laws are invariant under common nonnegative time shifts; call this strict stationarity. For $j\ge 0$, set ${\mathcal{F}}_j^{-}:=\sigma (Y_0,\dots ,Y_j)$ and ${\mathcal{F}}_j^{+}:=\sigma (Y_j,Y_{j+1},\dots )$. For $n\ge 1$, define the strong-mixing coefficient by ${\alpha }_Y(n):={\sup }_{j\ge 0}{\sup }_{\begin{array}{c}A\in {\mathcal{F}}_j^{-}\\ B\in {\mathcal{F}}_{j+n}^{+}\end{array}}\mathrm{\mid }\mathbb{P}(A\cap B)-\mathbb{P}(A)\mathbb{P}(B)\mathrm{\mid }\mathrm{.}$

Define the uniform-mixing coefficient by ${\varphi }_Y(n):={\sup }_{j\ge 0}{\sup }_{\begin{array}{c}A\in {\mathcal{F}}_j^{-}\\ \mathbb{P}(A)>0\end{array}}{\sup }_{B\in {\mathcal{F}}_{j+n}^{+}}\mathrm{\mid }\mathbb{P}(B\mid A)-\mathbb{P}(B)\mathrm{\mid }\mathrm{.}$

For square-integrable real random variables $U,V$ with positive variances, write $\mathrm{Corr}(U,V):=\frac{\mathrm{Cov}(U,V)}{\sqrt{\mathrm{Var}(U)\mathrm{Var}(V)}}\mathrm{.}$ For each $n\ge 1$, let ${\mathcal{C}}_n$ consist of the triples $(j,U,V)$ such that $j\ge 0$, $U,V$ are real and square-integrable, $U$ and $V$ are measurable with respect to ${\mathcal{F}}_j^{-}$ and ${\mathcal{F}}_{j+n}^{+}$ respectively, and both have positive variance. Define the maximal-correlation coefficient by ${\rho }_Y(n):=\sup (\{0\}\cup \{\mathrm{\mid }\mathrm{Corr}(U,V)\mathrm{\mid }:(j,U,V)\in {\mathcal{C}}_n\}),$ where square-integrability is in the sense of The space $L^p(\mu )$ as the quotient by null functions. Thus ${\rho }_Y(n)=0$ when there is no admissible pair, as for a constant process. The process is called strongly mixing, uniformly mixing, or rho-mixing, respectively, when the corresponding coefficient tends to zero. These coefficients are distinct conditions; no implication between them is included in this definition.