Title: Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction,
Abstract: In this paper, we consider time-inhomogeneous branching processes and
time-inhomogeneous birth-and-death processes, in which the offspring
distribution and birth and death rates (respectively) vary in time. A classical
result of branching processes states that in the critical regime, a process
conditioned on non-extinction and normalized will converge in distribution to a
standard exponential. In a paper of Jagers, time-inhomogeneous branching
processes are shown to exhibit this convergence as well. In this paper, the
hypotheses of Jagers' result are relaxed, further hypotheses are presented for
convergence in moments, and the result is extended to the continuous-time
analogue of time-inhomogeneous birth-and-death processes. In particular, the
new hypotheses suggest a simple characterization of the critical regime.