Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$). Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the condition of the generalized Perron-Frobenius theorem. In particular, there exists a Perron eigenvalue $\lambda >0$ (assumed to be finite) and a strictly positive eigenvector $x = (x_i)$ such that $Ax =\lambda x$.
My question: under what conditions on $A$ can one find a probability eigenvector $x$, i.e., $\sum_i x_i =1$?
I hope that this question has been discussed in the literature, and any references or suggestions are greatly appreciated.
Remark There is an obvious condition: $A$ is positive recurrent and has the equal column sum property $\sum_j a_{ij} = c, \forall i$. But this condition seems too strong.
Remark I missed to mention an important property of $A$: for all $n$, the entries $a^{(n)}_{ij}$ of $A^n$ are finite. It happens, for example, when every column of $A$ contains only finitely many nonzero entries.
