Martingale in Galtton-Watson
We note $m$ as the expectation of the production. That is$$
m = \sum_{k=1}^{\infty} k p_k
$$
A simple retira of extinction is that if $m \leq 1$, almost surely, the population will die. But if $m > 1$, the population has a probability to survive and this probability is the smaller fixed point of the characteristic function.
$$
q = \phi(q)
$$
If we would like to say more about the longtime behavior, in the sub-critical case, we have quasi-stationary theory, which is another story of my EA project. In the sup-critical case, we know that
$$
W_n = \frac{Z_n}{m^n}
$$
is a martingale, where $Z_n$ is the number of the population of each generation. The positivity means the convergence of martingale
$$
W_n \xrightarrow[p.s]{n \rightarrow \infty} W_{\infty}
$$
However, a natural question is in which case, we have U.I convergence or $L^1$ convergence. It has other sense in aspect of change of probability and we will study it later. Some special case is that when $\mathbb{E}(Z^2_1) < \infty$, then we have
$$
\begin{eqnarray*}
\mathbb{E}(W^2_{n+1}) & \leq &\mathbb{E}(W^2_{n}) + \frac{Var(Z_1)}{m^{n+2}} \\
\Rightarrow \sup_{n} \mathbb{E}(W^2_n) & < &\infty
\end{eqnarray*}
$$
Then we have a convergence in the sense $L^2$ and this implies the convergence in the sense $L^1$.
Generally, use the property of branch, we obtain that $\mathbb{P}(W_{\infty} = 0)$ is also a solution of $\phi(q) = q$. However, we have no idea that it is the smaller one. But in the case $\mathbb{E}(W_{\infty}) = 1$, we know that the in the sup-critical case it is not $1$. Then we get that
$$
\mathbb{P}(W_{\infty} = 0) = \mathbb{P}(\text{Extinction})
$$
so conditionally non-extinction, we an exponential increment, except that we don't know the constant.
Then exact equivalent condition of convergence will be discussed at last and we just discuss the Galton-Watson with immigration in the second part.
Galton-Watson with immigration
In the model of Galton-Watson with immigration, each step we have not only the reproduction but also the immigration of population. We can of course couple $Z_n$ with a classical Galton-Watson $X_n$, then whether $\frac{Z_n}{m^n}$ converge depends on the intrgration of $log(Y_1)$.
We skip the technique part, the result is that $$\begin{eqnarray*}\mathbb{E}(\log{Y_1}^+) < \infty \Rightarrow \lim_{n \rightarrow \infty} \frac{Z_n}{m^n} = c < \infty \\ \mathbb{E}(\log{Y_1}^+) = \infty \Rightarrow \lim_{n \rightarrow \infty} \frac{Z_n}{m^n} = \infty\end{eqnarray*}$$
Change of probability and Kesten-Stigum theorem
The Kesten-Stigum theorem is as following.
\begin{eqnarray}\mathbb{E}(W_{\infty}) &=& 1 \\
\mathbb{P}(W_{\infty} > 0 | \text{non-extinction}) &=& 1 \\
\mathbb{E}(Z_1 (\log{Z_1})^+) &<& \infty \\
\end{eqnarray}
The idea is very technique. To prove the $\mathbb{E}(W_{\infty}) = 1$, we transform the problem to the change of probability. Then it is a biased Galton-Watson model, which can also be considered as a model with immigration. The we apply the result of the model of immigration. Technique part need the decomposition of measure and the change of probability in filtration.
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Some remarks after two one years.
In 2017, I didn't understand all the part of this story, although the above gives almost all the points. The Kesten-Stigum theorem is important since it studies the the martingale of the Galton-Watson process. The critical and sub-critical case are easy : extinction. However, the super-critical case still has two cases : extinction or an exponential growth. The theorem tells us if there is no extinction, it is an exponential growth.
Secondly, to study the $W_n = \frac{Z_n}{m^n}$, we treat it as a change of probability i.e the probability $\mathbb{Q}$ of Galton-Watson process with immigration. We know that two measures can be decomposed into absolute continuous part and singular part. Moreover, in martingale case, it is that the part $\{W_{\infty} = \infty\}$ makes sense. The infinite part could not be seen as the part absolutely continuous, so we have to treat it specially.
Then the part of lemma Seneta is always technical. In fact, the number of individual of immigration contributes with a discount ration in the population. The criteria of $\log_+ Y$ often appears in the case of random environment.
