  In this paper the asymptotic behavior of a critical multi-type branching
process with immigration is described when the offspring mean matrix is
irreducible, in other words, when the process is indecomposable. It is proved
that sequences of appropriately scaled random step functions formed from
periodic subsequences of a critical indecomposable multi-type branching process
with immigration converge weakly towards a process supported by a ray
determined by the Perron vector of the offspring mean matrix. The types can be
partitioned into nonempty mutually disjoint subsets (according to communication
of types) such that the coordinate processes belonging to the same subset are
multiples of the same squared Bessel process, and the coordinate processes
belonging to different subsets are independent.
