Let $(N(t))_{t\geq 0}$ be a Galton-Watson process with reproduction law $p=(p_k)_{k\in\mathbb{N}}$ and rate of births $\tau >0$ i.e. we start with one individual and each individual in the system gives birth indepently with rate $\tau $ to a random number of offsprings distributed as $p$. If $p_0=0$ and $m : = \sum_{n\geq 1}np_n > 1$ then the stopping time $$T_n := \inf\{t > 0\colon N(t) > n\}$$ is finite a.e.
I want to prove that $$\frac{T_n}{\mathbb{E}T_n} \to 1 \text{ in }L^2$$ for this I think we can use dominated convergence and prove first that we have the convergence a.e. and then prove that $$\sup_n \frac{T_n}{\mathbb{E}T_n} \in L^2$$ But I don't know how to do either of those things.
Any help and idea will be appreciated
Thanks