Asymptotic Behavior of Multi-type Nearly Critical Galton–watson Processes with Immigration

where {ξn,j , εn: n, j ∈ N} are independent random variables with nonnegative integer values such that for each n ∈ N, {ξn,j : j ∈ N} are identically distributed. We can interpret Xn as the number of individuals in the n-th generation of a population, ξn,j is the number of offsprings produced by the j-th individual belonging to the (n− 1)-th generation, and εn is the number of immigrants in the n-th generation. A zero start one-dimensional inhomogeneous integer-valued autoregressive (INAR) time series is a special single-type GWI process, such that the offspring distributions are Bernoulli.