Limiting behavior of a generalized branching process with immigration

In the paper a modification of the branching process with continuous states, introduced by Adke and Gadag [1995. A new class of branching processes. Branching Processes. In: Proceedings of the First World Congress, Lecture Notes in Statistics, Springer, Berlin, vol. 99, pp. 90-105] is considered. Theorems establishing connection between asymptotic behavior of this process and the Bienaymé-Galton-Watson process are proved. Applications in the case of the process without and with stationary immigration are provided.