We compute exact values respectively bounds of "distances" - in the sense of
(transforms of) power divergences and relative entropy - between two
discrete-time Galton-Watson branching processes with immigration GWI for which
the offspring as well as the immigration is arbitrarily Poisson-distributed
(leading to arbitrary type of criticality). Implications for asymptotic
distinguishability behaviour in terms of contiguity and entire separation of
the involved GWI are given, too. Furthermore, we determine the corresponding
limit quantities for the context in which the two GWI converge to Feller-type
branching diffusion processes, as the time-lags between observations tend to
zero. Some applications to (static random environment like) Bayesian decision
making and Neyman-Pearson testing are presented as well.Comment: 45 page