Assume we have a multitype branching process, i.e., we have a mean matrix $M_{ij}$ and $M_{ij}$ is the expected count of generating $j$ from $i$ in one time step, i.e.:

$M_{ij} = \sum_{r} n(r,j)P(r | i)$

where $P(r | i)$ is the probability of choosing an action $r$ by type $i$ and $n(r,j)$ is the number of times $j$ is generated by this action.

Then we have $M^k$ to be the expected number of times $j$ will be generated by $i$ through $k$ time steps.

For $A = \sum_{k=0}^\infty M^k$, $A_{ij}$ would be the expected number of times that $j$ is generated at any time step by $i$.

Note that $A = (I-M)^{-1}$.

Does someone know where I can find a reference to the above statements, if they are indeed true? I tried looking all over for this in multitype branching processes literature, but couldn't find anything. I am basically looking for an interpretation, if exists, of $(I-M)^{-1}$ where $M$ is the mean matrix (momentum matrix) of a multitype branching process.