is there an interpretation to the inverse of $I-M$ in multitype branching process, where $M$ is the mean matrix?

Question

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.

Answer 1

In general, if $M$ was the transition matrix (infinitesimal generator) of a Markov chain , this functional would be called called the resolvent. Perhaps you already knew this; if not, you could look at James Norris' Markov Chains for a nice introduction; it may well contain enough information for you.

Answer 2

See Chapter V of the book "Branching Processes" by Athreya and Ney. In particular, in section 2 of that chapter is the interpretation of $M^k$ that you give above.

The sum $A=\sum_{k=0}^\infty M^k = (I-M)^{-1}$ is correct if $\rho < 1$, where $\rho$ is the maximal eigenvalue of the matrix $M$. As is shown in Chapter V, section 3 of Athreya and Ney's book, the maximal eigenvalue $\rho$ plays the role of the critical parameter for the multi-type branching process. That is, the multi-type branching process dies out with probability 1 if and only if $\rho\leq 1$.