Following setting:
Consider $\left\{0,1,2\right\}^{\mathbb{Z}}$. And on it the following dynamics: A site $x\in \mathbb{Z}$ represents a cell which can be 1, 2 or 0. A 1 becomes a 2 in the next time step, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its 2 neighbors is a 1.
Let $\eta_n(x)$ denote the state of the cell at location $x$ at time $n$.
Consider the measure space $(\left\{0,1,2\right\}^{\mathbb{Z}},\mathcal{B},P)$, where $\mathcal{B}$ is the Borel-$\sigma$-algebra generated by the cylinder sets of the form $$ \text{cyl}(y_{i}^{i+n})=\left\{x\in\left\{0,1,2\right\}^{\mathbb{Z}}: x_i=y_i,x_{i+1}=y_{i+1},\ldots,x_{i+n}=y_{i+n}\right\} $$ and $P$ is the product measure with $$ P(\text{cyl}(y_i^{i+n}))=\prod_{k=i}^{i+n}p(y_k),~~\text{with}\sum_{k=0}^{2}p(k)=1. $$
Suppose that the systems starts with $P$ giving each state the probability $1/3$.
Now I found the following theorem:
It is $$ \text{Prob}\left\{\eta_n(0)=1\right\}\sim\sqrt{2/(27\pi n)}, \text{where } \sim \text{ means approximately equal}. $$
Furthermore, because $\text{Prob}\left\{\eta_{n+1}(0)=2\right\}=\text{Prob}\left\{\eta_n(0)=1\right\}$ it follows that $$ \text{Prob}\left\{\eta_n(0)\neq 0\right\}\sim\sqrt{8/(27\pi n)}. $$
Now to my question:
The author says that by this it follows that $$ \eta_n\to 0\text{ in distribution as }n\to\infty. $$
And this is what I cannot understand.
The theorem says that eventually there will be a 0 at the cell at position 0. But why should there eventually be 0 at all cells?