Asymptotic behavior of critical indecomposable multi-type branching processes with immigration^∗

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720 Szeged, Hungary.
tivadar.danka@math.u-szeged.hu

Received: 13 May 2015
Revised: 9 December 2015
Accepted: 10 March 2016

Abstract

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.