Critical branching processes with immigration : scaling limits of zeroes and meanders

We study the hitting times of 0 and the probability of the local extinctions of Bienaymé- Galton-Watson processes with immigration (BGWI). When the tails of the immigration and regeneration distributions vary regularly, we establish the asymptotics for these probabilities, which yield a Yaglom limit, i.e., the limiting distribution of the process conditioned to stay (strictly) positive up to some positive time. As it is well known, the scaled Bienaymé-Galton-Watson processes with immigration converge to Continuous Branching Processes with immigration (CBI). In the setting of this scaling limit, we extend the one-dimensional Yaglom limit to a limit in Skorokhod’s space of càdlàg functions. This yields an invariance principle for meanders of CBIs. We put these all together to derive a scaling limit for the na¨ıve counting local time of the BGWI. Lastly, we show how to sample trajectories of the considered processes. This last chapter is made into a Python3 library, available by link.