Abstract:

Nerman (1981) showed that, under certain assumptions including the
existence of a Malthusian parameter $\alpha > 0$,
a supercritical general branching process $\mathcal{Z}_t^\phi$ counted with random characteristic $\phi$
rescaled by $e^{-\alpha t}$ converges as $t \to \infty$ to a constant (depending on $\phi$ and the reproduction point process $\xi$)
times the limit of Nerman's martingale $W \geq 0$.

The Malthusian parameter is described by the relation $\tilde \mu(\alpha)=1$ for $\alpha > 0$
where $\tilde \mu$ denotes the Laplace transform of the intensity measure $\mu = \mathbf{E}[\xi]$ of the reproduction point process $\xi$.
Associated with the Malthusian parameter is Nerman's martingale.
However, the equation $\tilde \mu(\lambda)=1$ may have further solutions in the complex plain
and associated with each solution is a complex Nerman-type martingale.

We establish an asymptotic expansion of $\mathcal{Z}_t^\phi$ as $t \to \infty$
with Nerman's martingale figuring in the leading-order term and further complex martingales
determining the lower-order terms. Only the roots $\tilde \mu(\lambda)=1$ with $\mathrm{Re}(\lambda) \geq \frac\alpha2$
appear in this expansion contributing terms of the order $e^{\lambda t}$.
Further, there are Gaussian fluctuations of the order of the square root of the population growth rate $e^{\alpha t}$.
Our results extend and sharpen earlier ones obtained by Janson (2018).