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

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 $\mathit{q}(A)$ of extinction in subsets of types $A\subseteq $\mathcal{X}$_d$maydifferfromtheglobalextinctionprobability$\boldsymbol{q}$andthepartialextinctionprobability$\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\mathit{q}<\mathit{q}(A)<\tilde{\mathit{q}}$. We then present an iterative method to compute the vector $\mathit{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\mathit{q}(A)$ in the set of fixed points of the progeny generating vector.