Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function

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 equal to 1. Under a growth condition on the variances of the offspring distributions, assuming the locally uniform convergence of the expectation functions of properly scaled immigration processes towards a càdlàg function, and the locally uniform convergence of the mean absolute deviation functions of the scaled immigration processes in question towards zero, we establish a functional limit theorem for the sequence of Galton-Watson processes with inhomogeneous immigration in question towards a deterministic limit process. We also provide some applications, among others, we show that a recent result of Rahimov and Sharipov (2024) on a functional limit theorem for a sequence of nearly critical Galton-Watson processes with dependent immigration can be recovered using our result.