Abstract We investigate branching processes in a nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely; therefore, we either condition on non-extinction or add inhomogeneous immigration. Extending our one-dimensional limit results from Kevei and Kubatovics (2024), we derive functional limit theorems. In the former case, the limit process is a time-changed simple birth-and-death process on (-, ) conditioned on survival at 0, while in the latter, it is a time-changed stationary continuous-time Markov branching process with immigration.
Kevei et al. (Wed,) studied this question.