P \text{-a.s.}$$ So $P$-almost surely for all $t$ sufficiently large $$|N_t^{\lambda t}| \leq |\hat{N}_{\lfloor t \rfloor}^{\lambda \lfloor t \rfloor}| \leq 
\mathrm{e}^{(\Delta_\lambda + \epsilon)\lfloor t \rfloor}$$ Then if $\lambda > \lambda_{crit}$ we can take $\epsilon$ sufficiently small so that $\Delta_\lambda + \epsilon < 0$ and hence $|N_t^{\lambda t}| < 1$ for $t$ large enough thus proving .

If $\lambda < \lambda_{crit}$ then we get $$\limsup_{t \to \infty} \frac{1}{t} \log |N_t^{\lambda t}| \leq \Delta_\lambda + \epsilon
\qquad P \text{-a.s.}$$ and letting $\epsilon \to 0$ yields .

Finally, if $\lambda > \lambda_{crit}$ then follows from Markov inequality and : $$\limsup_{t \to \infty} \frac{1}{t} \log P (|N_t^{\lambda t}| > 0) \leq 
\limsup_{t \to \infty} \frac{1}{t} \log E|N_t^{\lambda t}| = \Delta_{\lambda}.$$

Note that the $X \log X$ condition on the offspring distribution was not required so far. It will be essential in the next subsection.

Additive martingales and applications
-------------------------------------

Recall that under the probability $\tilde{P}$ the branching process together with the spine may be described as follows:

-   The process starts with a single spine particle whose path $(\xi_t)_{t \geq 0}$ is distributed like a Brownian motion.

-   At instantaneous rate $\beta_0 \delta_0(\cdot) + \beta$ along its path the spine particle splits into $A(\cdot)$ 