  This paper studies finite-time safety and reach-avoid verification for
stochastic discrete-time dynamical systems. The aim is to ascertain lower and
upper bounds of the probability that, within a predefined finite-time horizon,
a system starting from an initial state in a safe set will either exit the safe
set (safety verification) or reach a target set while remaining within the safe
set until the first encounter with the target (reach-avoid verification). We
introduce novel barrier-like sufficient conditions for characterizing these
bounds, which either complement existing ones or fill gaps. Finally, we
demonstrate the efficacy of these conditions on two examples.
