2 added 64 characters in body

Yes, condition uniqueness holds.

Condition 2 implies that every state $j$ is either absorbing $(j\not\in H)$ or transient $(j\in H)$. Define the absorption time to be $T=\inf (n\geq 0: X_n\not\in H)$. This $T$ is almost surely finite for any starting state $i$. i$, that is, the chain is eventually absorbed. If$j\in H$, then$p^n_{ij}=P_i(X_n=j)\leq P_i(T>n)\to 0=:Q_{ij}$as$n\to\infty$. If$j\not\in H$, then$p^n_{ij}=P_i(X_n=j)\uparrow P_i(X_T=j)=:Q_{ij}$as$n\to\infty$. 1 Yes, condition 2 implies that every state$j$is either absorbing$(j\not\in H)$or transient$(j\in H)$. Define the absorption time to be$T=\inf (n\geq 0: X_n\not\in H)$. This$T$is almost surely finite for any starting state$i$. If$j\in H$, then$p^n_{ij}=P_i(X_n=j)\leq P_i(T>n)\to 0=:Q_{ij}$as$n\to\infty$. If$j\not\in H$, then$p^n_{ij}=P_i(X_n=j)\uparrow P_i(X_T=j)=:Q_{ij}$as$n\to\infty\$.