I'm studying the following example but unable to understand a few things:
Assume all particles act independently of each other, and the probability that a particle produces $k$ offsprings is the same for all particles. Denote this probability with $f_k$, $k=0,1,2...,$ and the population size of the $t$-th generation by $X_t$, viewing $t = 0,1,...$ as time moment.
Goal is to calculate extinction probability. Let $u_t = P(X_t = 0)$ and observe that $u_t <= u_{t+1}$. So sequence ${u_t}$ is non-decreasing, all $u_t <= 1$ and hence, there is exists a number $u = \lim_{t \rightarrow \infty} u_t$ which we will call an extinction probability.
Suppose, for a while, that $X_0 = 1$ and consider the events $A$ = {the population will extinct} and $B_k$ = {the original particle has produced $k$ offsprings}, $k=1,2,...$
Since the particles act independently, once a particle has been born, the extinction probability for the part of the population generated only by this particle is the same $u$. If the original particle has produced exactly $k$ offsprings, the whole population will disappear if and only if all branches corresponding to the k original offsprings disappear. Hence, the extinction probability for the whole population is $u^k$. The formula for total probability
$u = P(A) = \sum_{k=0}^\infty P(A|B_k)P(B_k) = \sum_{k=0}^\infty u^kf_k = \sum_{k=0}^\infty f_ku^k$
Question 1:
I don't quite understand how $u$ appear on the left hand side of the expression above and on the right hand side of it.
Question 2:
I'm confused by the statement that: once a particle has been born, the extinction probability for the part of the population generated only by this particle is the same u...Hence, the extinction probability for the whole population is $u^k$. My confusion stems from if $u_t = P(X_t = 0)$ where $u_t$ is the probability of extinction of whole population at time t and $u = \lim_{t \rightarrow \infty} u_t$, then how does $u_t$ relate to $u^k$.
