It follows from the Borel-Cantelli Lemma that $$P \Big( \Big\{ \frac{1}{n} \log |N_n| > \frac{1}{2}\hat{\beta}_0^2 + \hat{\beta} + \epsilon \Big\} \text{ i.o. }\Big) = 0$$ and thus $$\limsup_{n \to \infty} \frac{1}{n} \log |N_n| \leq \frac{1}{2} \hat{\beta}_0^2 + \hat{\beta} + 
\epsilon \qquad P \text{-a.s.}$$ By letting $\epsilon \to 0$ we establish with the limit taken over integer times. To get convergence over any real-valued sequence we note that $(|N_t|)_{t \geq 0}$ is a non-decreasing process and so for any $t > 0$ $$\frac{1}{t}\log |N_t| \leq \frac{\lceil t \rceil}{t} \frac{\log |N_{\lceil t \rceil}|}{\lceil t \rceil}$$ and hence $$\limsup_{t \to \infty} \frac{1}{t} \log |N_t| \leq \limsup_{t \to \infty} 
\frac{\log |N_{\lceil t \rceil}|}{\lceil t \rceil}
\leq \frac{1}{2} \hat{\beta}_0^2 + \hat{\beta} \qquad P \text{-a.s.}$$

For upper bounds of Lemma \[pop\_lambda\_as\] we need to adjust the previous argument because unlike $(|N_t|)_{t \geq 0}$ the process $(|N_t^{\lambda t}|)_{t \geq 0}$ is not monotone. We first establish the following result.

\[lambda\_sup\] For $\lambda > 0$ and $n \in \mathbb{N}\cup\{0\}$ we define the following set of particles: $$\hat{N}^{\lambda n}_n := \big\{ u \in N_{n+1} \ : \ \sup_{s \in [n, n+1]}X^u_s \geq \lambda n\big\}.$$ Then $$\limsup_{n \to \infty} \frac{1}{n} \log E |\hat{N}^{\lambda n}_n| \leq \Delta_\lambda.$$

Note that for any 