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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.

Community
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Salamo
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1 Answers1

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Now I really wonder what is meant with X, what exactly is X?

The random variable $X$ is any random variable distributed like $X_{1,1}$. It appears only through its distribution.

Did
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  • Which text are you following? – Did Nov 06 '14 at 18:58
  • I only use my papers I wrote in the lecture. – Salamo Nov 06 '14 at 19:13
  • Could you please explain to me, how one can read from the task that $X$ is distributed like $X_{1,1}$? – Salamo Nov 06 '14 at 19:13
  • Is it possible that it is NOT POSSIBLE that $q=0$? – Salamo Nov 06 '14 at 19:35
  • "Could you please explain to me, how one can read from the task that X is distributed like X1,1?" One cannot, the "task" is not very well written (but I know this setting, which is why I can add what is missing). – Did Nov 06 '14 at 19:57
  • "Is it possible that it is NOT POSSIBLE that q=0?" No, in some cases one has q=0. – Did Nov 06 '14 at 19:58
  • My idea is that $P(\exists n\in\mathbb{N}: N_n=0)=P(N_1=0)=P(X=0)=1-P(X>0)=1-g_X(1)$ where $g_X$ is the generating function. And it is $g_X(1)=P(X<\infty)=1$. But what is now the condition that is asked? – Salamo Nov 06 '14 at 20:49
  • ?? But P(∃n∈ℕ:Nn=0) is not P(N1=0), is it? – Did Nov 06 '14 at 20:55
  • You are right. Thats wrong. How to start then? – Salamo Nov 06 '14 at 21:01
  • Is this your question? I thought your question was to understand X. – Did Nov 06 '14 at 21:03
  • Yes, my original question was how to understand $X$. But now this new question arose. I wonder if I should use $q=1-P(\forall n\in\mathbb{N}: N_n>0)$ and to continue with that; whereat my question would be: how to do so. – Salamo Nov 06 '14 at 21:07
  • New problem $\implies$ new question. – Did Nov 06 '14 at 21:08
  • Ok, I will post a new thread for m new question! Thank you. – Salamo Nov 06 '14 at 21:09