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In a Galton-Watson branching process the extinction probability is sometimes given by $$\lim_{t \rightarrow \infty} P[X(t)=0]$$ and sometimes as $$ P[\lim_{t \rightarrow \infty}X(t)=0]$$ Is there a difference between These two formulations? Which is the "correct" one, if yes?

Could you give me a link to a paper where this is written?

Here is the Wikipedia page, but unfortunately, the linked papers there do not help..

Thank you for your help

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  • $\begingroup$ In general, there is a difference between the two formulations. Unfortunately, I'm not familiar with Galton-Watson processes... $\endgroup$ – saz Aug 16 '14 at 10:16
  • $\begingroup$ Literally, the former means the limit of a sequence of real numbers; while the latter means the probability of the convergence of a given sequence. So there should be a difference. $\endgroup$ – Megadeth Aug 16 '14 at 10:18
  • $\begingroup$ I see that this Argument implies that it can't be the same exactly, as These are two different formulations, the one a squence of random variables, the other a sequence of probabilities. What I wonder whether the two Limits are then the same or not.. Does this also hold, that this is in General not the same? $\endgroup$ – user146358 Aug 16 '14 at 10:27
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In the context of branching processes, the two statements are indeed equivalent. This is due to the following facts:

  • The value of each random variable $X(t)$ is almost surely a nonnegative integer.
  • The events $[X(t)=0]$ are nondecreasing with respect to $t$.

Thus, $P(X(t)=0)\to P(A)$ when $t\to\infty$, where $A=\bigcup\limits_t[X(t)=0]$. On the other hand, the event $[\lim\limits_{t\to\infty}X(t)=0]$ is also $[\exists t,X(t)=0]=A$, QED.

To sum up, the reason why the assertion holds is that, if $x:t\mapsto x(t)$ is an integer valued function such that if $x(t)=0$ then $x(s)=0$ for every $s\geqslant t$, then $x(t)\to0$ when $t\to+\infty$ if and only if $x(t)=0$ for some $t$.

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  • $\begingroup$ thank you for your answer. that helps a lot. Do you know a link to a paper, where the mathematical Definition of a Galton-Watson process is stated? All I can find is Wikipedia and the references there do not lead to formal definitions, either. $\endgroup$ – user146358 Aug 16 '14 at 12:57
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    $\begingroup$ The WP page has a formal definition. Some recent introductory notes on the subject called "Probability on Trees: An Introductory Climb" are available on Yuval Peres webpage. $\endgroup$ – Did Aug 16 '14 at 13:03
  • $\begingroup$ I know that Wikipedia has a formal Definition, and this is enough for me. But the Problem is, that I am not allowed to reference to Wikipedia. Thank you. $\endgroup$ – user146358 Aug 16 '14 at 13:05
  • $\begingroup$ Quote: "the references there do not lead to formal definitions". $\endgroup$ – Did Aug 17 '14 at 9:20

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