The probabilities of extinction in a branching random walk on a strip

Abstract

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set ${\mathcal{X}}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\mathbf{q}(A)$ of extinction in subsets of types $A\subseteq {\mathcal{X}}_d$ may differ from the global extinction probability $\mathbf{q}$ and the partial extinction probability $\tilde{\mathbf{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\mathbf{q}<\mathbf{q}(A)<\tilde{\mathbf{q}}$. We then present an iterative method to compute the vector $\mathbf{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\mathbf{q}(A)$ in the set of fixed points of the progeny generating vector.