Let $X_n$ be a (could assume this is homogenous) markov chain on a countable state space $S$, and write $N_n(x)=\sum_{k=1}^n 1_{\{X_k=x\}}$. Let $z\in S$ be a recurrent state, denote by $R_z$ its accessibility equivalence class, and let $\mu_z(y)=\mathbb{E}_zN_{\tau_z}(y)=\mathbb{E}(N_{\tau_z}(y)|X_0=z)$ where $\tau_z=\inf\{k: X_k=z\}$. Prove that $$\lim_{n\rightarrow\infty} \frac{N_n(y)}{N_n(z)}=\mu_z(y)$$ holds $\mathbb{P}_x$ a.s for $x,y\in R_z$ where $\mathbb{P}_x(X)=\mathbb{P}(X|X_0=x)$.
Writing $\mu_z$ out, this is similar to Laws of large numbers? But I didn't get very far from it. Any hints, idea? Thanks.