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.