  We consider a general discrete-time branching random walk on a countable set
X. We relate local, strong local and global survival with suitable inequalities
involving the first-moment matrix M of the process. In particular we prove
that, while the local behavior is characterized by M, the global behavior
cannot be completely described in terms of properties involving M alone.
Moreover we show that locally surviving branching random walks can be
approximated by sequences of spatially confined and stochastically dominated
branching random walks which eventually survive locally if the (possibly
finite) state space is large enough. An analogous result can be achieved by
approximating a branching random walk by a sequence of multitype contact
processes and allowing a sufficiently large number of particles per site. We
compare these results with the ones obtained in the continuous-time case and we
give some examples and counterexamples.
