Show that $\left(V_{n}\right)_{n \geq 1}$ converges in $L_{1}$ Let consider the Galton-Watson process with immigration which is given by the following recursion
$$
Z_{n+1}=\sum_{k=1}^{Z_{n}} \xi_{k}^{(n+1)}+\eta_{n+1}
$$
where $\left(\xi_{k}^{(n)}\right)_{k \geq 1, n \geq 1}$ are i.i.d. copies of $\xi$ and $\left(\eta_{n}\right)_{n \geq 1}$ are i.i.d. copies of $\eta$ such that $\xi$ and $\eta$ take values in $\mathbb{Z}_{+}=\{0,1,2, \ldots\}$ and $m:=\mathbb{E}[\xi]>1, \lambda:=\mathbb{E}[\eta] \in(0, \infty)$. Suppose also that $\left(\xi_{k}^{(n)}\right)_{k \geq 1, n \geq 1}$ are independent of $\left(\eta_{n}\right)_{n \geq 1}$.
$\text { Set } V_{n}=Z_{n} / m^{n}$ Show that $\left(V_{n}\right)_{n \geq 1}$ converges in $L_{1}$ i.e.
$$
\lim _{n \rightarrow \infty} \mathbb{E}\left[\left|V_{n}-V_{\infty}\right|\right]=0 \text {, }
$$
for some $V_{\infty}$, and conclude that $\mathbb{P}\left(V_{\infty}>0\right)>0$
My attempt
To show that $V_n$ converges in $L_{1}$, can I use the fact that since $V_n$ is a martingale which means it is a sub-martingale and super-martingale. Therefore:
$$ E[|V_0|] \leq E[|V_n|] \leq E[|V_0|] $$
$$ 1 \leq E[|V_n|] \leq 1 $$
$$ E[|V_n|] = 1 \leq \infty  $$
Hence $V_n$ converges in $L_{1}$ Am I allowed to do this?
How do I got about showing the probability?
Thank you!
 A: The almost sure convergence to a finite limit is a Theorem due to Seneta, for which an elegant proof was found by Asmussen and Hering. The assumption that $E(\eta)<\infty$ can be relaxed to $E(\log_+ \eta)<\infty$. See Theorem 3.1 page 1129 n [1] for a self-contained presentation of this proof. For positivity of the limit and the $L_1$ convergence a further condition is needed,
namely the Kesten-Stigum condition $E(\xi \log \xi)<\infty$. Without this condition, the a.s. limit is 0, so $L_1$ convergence fails. See [1] or [2].
If one assumes the Kesten-Stigum condition, then  to prove the L^1 convergence, combine the argument in the proof noted above with the proof of the
Kesten-Stigum Theorem in [1] or [2].
[1] Lyons, Russell, Robin Pemantle, and Yuval Peres. "Conceptual proofs of L log L criteria for mean behavior of branching processes." The Annals of Probability (1995): 1125-1138.
https://www.jstor.org/stable/pdf/2244865.pdf
[2] Kesten, H. and Stigum, B.P., 1966. A limit theorem for multidimensional Galton-Watson processes. The Annals of Mathematical Statistics, 37(5), pp.1211-1223.
