in expectation from the previous subsection with the almost sure convergences.

For all the results in this subsection we shall have to impose an additional condition on the offspring distribution commonly known as the $X \log X$ condition (for some discussion see e.g. [@K04], [@Lyons] or [@LYS13]): $$\label{xlogx}
\sum_{n \geq 1} p_n n \log n < \infty \text{ and } \sum_{n \geq 1} q_n n \log n < \infty \text{.}$$ This condition is needed to ensure that certain martingales have non-zero limits as we shall see in Subsection 2.3.

Our first result, which should be compared with , is the almost sure approximation of the population size.

\[population\_as\] Suppose that condition on the offspring distribution is satisfied. Then $$\label{eq_pop_as}
\lim_{t \to \infty} \frac{1}{t} \log |N_t| = \frac{1}{2} \hat{\beta}_0^2 + \hat{\beta} \qquad P \text{-a.s.}$$

Next, unarguably the most important result of this paper, is the almost sure approximation of $|N_t^{\lambda t}|$ and it should be compared with .

\[pop\_lambda\_as\] Suppose that condition is satisfied. Take any $\lambda > 0$ and let $\Delta_\lambda$ be as in and $\lambda_{crit}$ as in .

If $\lambda < \lambda_{crit}$ then $$\label{subcrit}
\lim_{t \to \infty} \frac{1}{t} \log |N_t^{\lambda t}| = \Delta_\lambda \ (> 0) \qquad P \text{-a.s.}$$

