Two-sided stationary extension preserves maximal-correlation bounds

For finite $F\subset \mathbb{Z}$, choose an integer $m\ge 0$ with $F+m\subset {\mathbb{N}}_0$ and prescribe the law of $(X_i{)}_{i\in F}$ to be that of $(Y_{i+m}{)}_{i\in F}$. If $m^{\mathrm{\prime }}$ is another permissible shift, stationarity compares the smaller shift with the larger one, so the law is independent of the choice. Restrictions are consistent because finitely many coordinate sets can be translated by one common shift. Every coordinate space is the standard Borel space $\mathbb{R}$, so [F1] gives $Q$. Its nonnegative restriction equals the process law of $Y$ on cylinders and hence everywhere by [F5]. Translating finite sets does not change their prescribed laws, so $Q\circ T^{-1}=Q$ by the same uniqueness argument. Both $T$ and its inverse are measurable because their coordinate maps are coordinates.

For any subset $I\subset \mathbb{Z}$, let ${\mathcal{A}}_I$ be the algebra of events depending on finitely many coordinates in $I$. Every event in $\sigma (X_i:i\in I)$ can be approximated in symmetric-difference probability by ${\mathcal{A}}_I$. Indeed, the approximable events are closed under complements. For a countable union, first replace that union by a finite subunion with arbitrarily small lost probability, using continuity from below, then approximate its finitely many constituent events and apply the union bound. Thus the approximable events form a sigma-algebra containing ${\mathcal{A}}_I$. By [F4], approximate an $L^2$ variable by simple functions, and approximate the finitely many indicator events of each simple function by ${\mathcal{A}}_I$ events. For an indicator the squared $L^2$ error is its symmetric-difference probability; finite summation gives the required arbitrarily small $L^2$ error for a simple function. Consequently every such variable has finite-coordinate $L^2$ approximants.

First suppose $U$ and $V$ depend on finitely many coordinates in $(-\mathrm{\infty },k]$ and $[k+n,\mathrm{\infty })$, respectively. Translate all coordinates by one integer $m\ge 0$ so that the used past coordinates and the split $k+m$ are nonnegative. Their joint distribution is an original-process distribution by step 1.1. The translated variables are measurable in $\sigma (Y_0,\dots ,Y_{k+m})$ and $\sigma (Y_{k+m+n},Y_{k+m+n+1},\dots )$, so [F2] proves the asserted covariance inequality when both variances are positive. A variable of variance zero equals its mean almost surely and has zero covariance, so the inequality holds then too. For arbitrary $U,V$, take step 1.2's finite-coordinate $L^2$ approximants. Their means, second moments, and covariances converge by [F3], so passage to the limit proves the same inequality.

Assume ${\rho }_Y(n)\to 0$, and let $A$ satisfy $Q(A\mathrm{△}{T}^{-1}A)=0$, with $p=Q(A)$. Given $\epsilon >0$, step 1.2 supplies a finite-coordinate event $C$ with $Q(A\mathrm{△}C)<\epsilon$; enlarge its coordinate support to $[-m,m]$ for some $m\ge 0$. For $n>2m$, $C$ is in ${\mathcal{F}}_m$ and $T^{-n}C$ is in ${\mathcal{G}}_{n-m}$. Step 2.1 applied to their indicators gives $\mathrm{\mid }Q(C\cap T^{-n}C)-Q(C{)}^2\mathrm{\mid }\le {\rho }_Y(n-2m)\mathrm{/}4$, since every indicator variance is at most $1\mathrm{/}4$. Invariance modulo null sets and measure preservation imply $Q(A\cap T^{-n}A)=p$. The intersection changes in probability by at most $2\epsilon$ upon replacing both copies of $A$ by copies of $C$, and $\mathrm{\mid }{p}^2-Q(C{)}^2\mathrm{\mid }\le 2\epsilon$. Hence $\mathrm{\mid }p-p^2\mathrm{\mid }\le 4\epsilon +{\rho }_Y(n-2m)\mathrm{/}4$. Let $n\to \mathrm{\infty }$ and then $\epsilon \downarrow 0$ to obtain $p=p^2$. Therefore $p\in \{0,1\}$, proving [F6] and the stronger mod-null formulation claimed.