AbstractWe 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, we study the probabilities
$\textbf{\textit{q}}(A)$
of extinction in sets of types
$A\subseteq \mathcal{X}_d$
. We compare
$\textbf{\textit{q}}(A)$
with the global extinction probability
$\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$
, that is, the probability that the population eventually becomes empty, and with the partial extinction probability
$\tilde{\textbf{\textit{q}}}$
, that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for
$\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$
. We then present an iterative method to compute the vector
$\textbf{\textit{q}}(A)$
for any set A. Finally, we investigate the location of the vectors
$\textbf{\textit{q}}(A)$
in the set of fixed points of the progeny generating vector.
Random additions in urns of integers
