Long-Term Behavior of Subordinated Branching Processes with Prevailing Emigration

This paper deals into the long-term behavior of subordinated critical branching processes with migration. We focus on scenarios where emigration is the dominant factor and introduce additional randomness in timing through a subordination mechanism, involving renewal processes. The key findings highlight how the initial population size and the interarrival mean time influence both asymptotic behavior of the non-extinction probability and corresponding Yaglom type limit theorems. We also study an alternating regenerative process, when the population cycles between zero and positive states. This research complements previous studies for processes when immigration prevails over emigration.