Functional limit theorems for Galton--Watson processes with inhomogeneous immigration

We study the asymptotic behavior of a sequence of Galton–Watson processes with inhomogeneous immigration when the limit of the means of the offspring distributions is less than $1$ or equal to $1$. Under growth conditions on the expected values of the immigration distributions and the variances of the offspring distributions, and assuming the weak convergence of properly scaled immigration processes towards a non-negative stochastic process $\mathcal{Y}$ with càdlàg or continuous sample paths, we establish functional limit theorems for the sequence of Galton–Watson processes with inhomogeneous immigration in question. The limit stochastic processes can be represented as a constant multiple or an integral functional of $\mathcal{Y}$.