Consider a population such that each member, independently from other members, at a certain instant of time is replaced by its offspring. Lets denote with $X_n$ $({n\ge 1})$ the amount of the population at the the $n$-th generation, and let $X_0$ be the starting population. If $$T_0=\inf\{n\in\mathbb N\,:\, X_n=0\},\quad(\inf({\emptyset})=+\infty)$$ The probability that the population dies, given $X_0=1$, is: $$\alpha:=P(T_0<\infty\,|\,X_0=1)$$
Now reading the book "Bosq, Nguyen - A Course in Stochastic", I don't understand a passage at page 71:
Since the subset of states $\{1,2,\ldots\}$ is a transient class (I understand this), we see that: $$\lim_{n\to\infty} X_n(\omega) =\left\{\begin{array} {ll} 0 & \textrm{if}\; T_0(\omega)<\infty\\ \infty & \textrm{if}\; T_0(\omega)=\infty\\ \end{array}\right.$$
Since $0$ is an absorbing state, the first line of the equation is clear, but why if the population does not die, then it explodes? Why an oscillating behavior can't occur?
Thanks in advance.
There can be no oscillations, because then there would be a state that you visit infinitely many times. Each time you visit that state, there is a positive probability for the process to go extinct. For example, just the probability of having no children in the next generation. Denote this probability $p > 0$. Since you visit this state infinitely many times, the probability to $not$ go extinct will be $(1-p)^{\infty} = 0.$