Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?


share|improve this question

1 Answer 1

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.