Extinguishing behaviors for continuous-state nonlinear branching processes

Abstract

A nonnegative continuous-state branching process becomes extinguishing if it converges to 0 as time goes to infinity but never hits 0 in finite time. We consider a class of continuous-state nonlinear branching processes obtained from spectrally positive stable like Lévy processes by Lamperti type time changes using regularly varying (at 0+) rate functions, and obtain several large time asymptotic results on the extinguishing behaviors. In particular, we show that, depending on whether the stable index for the spectrally positive Lévy process is smaller than or equal to the regularly varying index for the rate function, a phase transition occurs from convergence in distribution to convergence in probability for the rescaled first passage times of levels approaching 0. For the later case, we further find conditions on almost sure convergence for the rescaled first passage times. We also obtain integral tests on almost sure long time fluctuations of the running minimum for the nonlinear branching process.