Let $X, X_{n,k}$ for $k,n\in\mathbb{N}$ denote independent random variables with values in $\mathbb{N}_0$. Define $N_0:=1$ and for $n\in\mathbb{N}$ set $$ N_n:=\begin{cases}0, & \text{ if }N_{n-1}=0\\X_{n,1}+\cdots+X_{n,N_{n-1}}, & \text{ if }N_{n-1}>0.\end{cases} $$ Find conditions on the distribution of $X$ for which the probability $$ q:=P(\exists n\in\mathbb{N}: N_n=0) $$ satisfies $q=0$.
In a first step I showed that $(N_n)_{n\in\mathbb{N}_0}$ is a Markov chain, see here (Check if $(N_n)$ is a Markov chain).
Now I really wonder what is meant with $X$, what exactly is $X$? I did not understand this yet.
Maybe you can help me?
If I undestood what $X$ shall be, maybe I then can find the asked condition.
Because we recently had generating functions maybe it then has sth to do with that.
The random variable $X$ is any random variable distributed like $X_{1,1}$. It appears only through its distribution.