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

Tivadar Danka, Gyula Pap
2016 E S A I M: Probability & Statistics  
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
more » ... 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. 1991 Mathematics Subject Classification. 60J80, 60F17, 60J60. The dates will be set by the publisher. TITLE WILL BE SET BY THE PUBLISHER with X (n) t := n −1 X nt for t ∈ R + , n ∈ N, where R + denotes the set of non-negative real numbers, x denotes the integer part of x ∈ R, and (X t ) t∈R+ is the pathwise unique strong solution of the stochastic differential equation (SDE) with initial value X 0 = 0, where m ε := E(ε 1 ), V ξ := Var(ξ 1,1 ), (W t ) t∈R+ is a standard Wiener process, and x + denotes the positive part of x ∈ R. We will investigate a p-type branching process (X k ) k∈Z+ with immigration. For simplicity, we suppose that the initial value is X 0 = 0. For each k ∈ Z + and i ∈ {1, . . . , p}, the number of individuals of type i in the k th generation is denoted by X k,i . The non-negative integer-valued random variable ξ k,j,i, denotes the number of type offsprings produced by the j th individual who is of type i belonging to the (k − 1) th generation. The number of type i immigrants in the k th generation will be denoted by ε k,i . Consider the random vectors
doi:10.1051/ps/2016010 fatcat:rp4sr37amvfinm2fthaecwme4a