Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ($n,m\in\mathbb{Z}$) and define $\mathcal{G}_n^m$ similarly for $Y_i$. Let $a$ be some fixed state of $X$ and consider the random variable

$$b_k:=\mathbb{P}[X_{2k+1}=a\mid \mathcal{F}_{0}^{2k}\vee\mathcal{G}_{0}^{k}]$$ and $$A_N:=\frac{1}{N}\sum_{k=1}^Nb_k$$

Question: Does $A_N$ converge almost surely?


To be more precise, let me frame this question in an ergodic-theoretic setting. Let $X$ and $Y$ be finite sets, $\Omega:=(X\times Y)^\mathbb{Z}$ and $\left(\Omega,\mu,T\right)$ be a measure-preserving system that is a Kolmogorov automorphism. Let $\pi_X:X\times Y\to X$ be the projection to $X$ and $X_i:\Omega\to X$ ($i\in\mathbb{Z}$) be the measurable function $X_i(\omega)=\pi_X(\omega_i)$ (similarly, $Y_i(\omega)=\pi_Y(\omega_i)$). Given a $\sigma$-algebra $\mathcal{H}$, we write $L^p(\mathcal{H})$ for the space of $\mathcal{H}$-measurable functions in $L^p$, and we also write $\mathcal{F}_n^m:=\sigma(\{X_i:\ n\leq i\leq m\})$, $\mathcal{G}_n^m:=\sigma(\{Y_i:\ n\leq i\leq m\})$, and, for $k\geq 0$, $P_k:L^1\to L^1$ for the orthogonal projection onto $L^1(\mathcal{F}_{-2k}^0\vee\mathcal{G}_{-2k}^{-k})$ (i.e., $P_k(f)=\mathbb{E}[f\mid\mathcal{F}_{-2k}^0\vee\mathcal{G}_{-2k}^{-k}]$). Fix some $a\in X$, write $A=\{\omega:\ X_1(\omega)=a\}$ and consider $$a_k:=P_k(1_A)$$ and $$A_N:=\frac{1}{N}\sum_{k=1}^NT^{2k}a_k$$

By invariance, both definitions of $A_N$ coincide

If $P_k$ were a monotone sequence of projections, then the affirmative answer would be trivial, since $a_k$ would be a pointwise converging martingale and, by Maker's generalization of Birkhoff's ergodic theorem, pointwise convergence of $a_k$ suffices.

In fact, it is true $A_N$ converges in $L^2$ (to $\int 1_A$) since one can prove that $a_k$ converges in $L^2$ to $\mathbb{E}[1_A\mid \mathcal{F}^0_{-\infty}]$ by showing there are monotone sequences of orthogonal projections $S_k$, $R_k$ (the first nondecreasing, the second nonincreasing) such that $S_k\leq P_k\leq R_k$ in the lattice of projections and $S_k\nearrow P$, $R_k\searrow P$, where $P$ projects onto $L^2(\mathcal{F}^0_{-\infty})$. One then uses Maker's theorem in the $L^2$ version.

It is also true that when $A_N$ is regarded as a Markovian operator $A_N(f)=(1/N)\sum_k^NT^{2k}P_k(f)$ then there is a dense subset $\mathcal{D}$ of $L^2$ such that $A_N(f)$ converges a.s. for any $f\in\mathcal{D}$. I would then like to use Banach's Principle to prove pointwise convergence on the whole of $L^2$ (which implies that of $A_N(1_A)$) but I can't prove the necessary bound $\sup_N |A_N(f)|<\infty$ for any $f\in L^2$. Any help here would be welcome.


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