Let $(\eta_{t}^{\rho})_{t\geq 0}$ be a totally asymmetric simple exclusion process (TASEP) from an initial configuration distributed according to the Bernoulli measure $\nu_{\rho}$ on $\{0,1\}^{\mathbb{Z}}$ ($\rho\in [0,1]$ constant).
It is well known that $\nu_{\rho}$ is ergodic for the process. Thus, by the Birkhoff ergodic theorem, we have that for every $f:\{0,1\}^{\mathbb{Z}}\longrightarrow\mathbb{R}$ depending on finitely many coordinates, $\lim\limits_{t\longrightarrow +\infty}t^{-1}\int\limits_{0}^{t}f(\eta^{\rho}_{s})\;ds=\nu_{\rho}(f)$ almost surely
My problem is the following: i need to prove the Rost's Theorem for the process $(\eta_{t}^{\rho})_{t\geq 0}$ (i presume that this case is some easy because our process is ergodic), that is, i need to prove that for all $a<b$ in $\mathbb{R}$, we have
$\lim\limits_{t\longrightarrow +\infty}t^{-1}\displaystyle\sum\limits_{x\in\mathbb{Z}:\;at\leq x\leq bt}\eta_{t}^{\rho}(x)=\rho(b-a)$ almost surely
(I am reading the last paper of Pablo Ferrari and he uses this result in the proposition 7.1, without mentioning: https://arxiv.org/pdf/1601.05346.pdf)
I tried to apply the Birkhoff ergodic theorem, but i failed at that attempt.
(My mathematics level is undergraduate, and this question arose working on my thesis)
