Growth scales and uniform integrability of branching processes in varying environments

We derive necessary and sufficient conditions for a sequence of constants $(C_{n})_{n \in \mathbb{N}_0}$ to be a growth scale of a branching process in (possibly defective) varying environments $(Z_{n})_{n \in \mathbb{N}_0}$ in the sense that $(Z_{n}/C_{n})$ is bounded from above and below with positive probability.

Along the way, we derive a Kesten-Stigum type result - necessary and sufficient conditions for branching processes with varying environments to converge in $L^1$ when normalised by their mean.

The proofs exploit truncation and change of measure arguments, convergence of random series and martingale techniques.