what is the link in between the mean matrix and also buildings of a branching procedure?

Allow $M$ be the mean matrix of a multitype branching procedure.

I am attempting to identify whether $A = (I-M)^{-1}$ has any kind of value.

I assume $A_{ij}$ would certainly represent the variety of times that type $j$ would certainly be created from type $j$, yet I can not locate any kind of product on that particular in the internet. Any person recognizes anything concerning that?

Many thanks.

2019-05-18 21:56:10
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Answers: 1

The matrix $M$ has $(i,j)$th access $M_{ij}=E(Z^j_1 | Z_0=e_i)$, that is, $M_{ij}$ is the mean variety of type $j$ spawn beginning with one type $i$ person.

In a similar way, 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 (overlooking concerns of merging), that $A_{ij}=E(\sum_{n\geq 0} Z^j_n | Z_0=e_i)$, the mean variety of type $j$ offspring beginning with a solitary type $i$ person.

An excellent reference is Harris is publication The Theory of Branching Processes .

2019-05-21 07:26:28