If $\lambda > \lambda_{crit}$ then $$\label{supercrit1}
\lim_{t \to \infty} |N_t^{\lambda t}| = 0 \qquad P \text{-a.s.}$$

and furthermore $$\label{supercrit2}
\lim_{t \to \infty} \frac{1}{t} \log P (|N_t^{\lambda t}| > 0) = \Delta_{\lambda} \ (< 0) \text{.}$$

As a direct corollary of Lemma \[pop\_lambda\_as\] we establish the almost sure speed of the rightmost particle.

\[cor\_rightmost\] Suppose that condition is satisfied. Then $$\label{rightmost}
\lim_{t \to \infty} \frac{R_t}{t} = \lambda_{crit} \qquad P \text{-a.s.,}$$ where $\lambda_{crit}$ is given in .

Outline of the paper
--------------------

The rest of this article is organised as follows. In Subsection 2.1 we follow the standard procedure of extending the probability space by constructing the spine process over our branching system. Then in Subsection 2.2 we recall the Many-to-One formula for branching processes and apply it to prove Proposition \[prop\_enx\] as well as the upper bounds for Lemma \[population\_as\] and Lemma \[pop\_lambda\_as\]. In Subsection 2.3 we construct certain change-of-mesure martingales and use them to prove lower bound for Lemma \[population\_as\] as well as “preliminary" lower bounds for Lemma \[pop\_lambda\_as\]. In subsections 3.1 and 3.2 we give the heuristic argument and the formal proof for lower bounds in Lemma \[pop\_lambda\_as\]. Finally, we conclude the paper with the proof of Corollary \[cor\_rightmost\].

