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Let $M$ be the mean matrix of a multitype branching process.

I am trying to figure out whether $A = (I-M)^{-1}$ has any significance.

I think $A_{ij}$ would stand for the number of times that type $j$ would be generated from type $j$, but I can't find any material on that in the web. Anyone knows anything about that?


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The matrix $M$ has $(i,j)$th entry $M_{ij}=E(Z^j_1 | Z_0=e_i)$, that is, $M_{ij}$ is the mean number of type $j$ offspring starting with one type $i$ individual.

Similarly, for the $n$th power of $M$ we get $M_{ij}^n=E(Z^j_n | Z_0=e_i)$

Expanding $A=(I-M)^{-1}=\sum_{n\geq 0} M^n$, we get (ignoring issues of convergence), that $A_{ij}=E(\sum_{n\geq 0} Z^j_n | Z_0=e_i)$, the mean number of type $j$ descendants starting with a single type $i$ individual.

A good reference is Harris's book The Theory of Branching Processes.

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