# Difference between $\lim P[…]$ and $P[ \lim ]$

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

• In general, there is a difference between the two formulations. Unfortunately, I'm not familiar with Galton-Watson processes... – saz Aug 16 '14 at 10:16
• 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. – Megadeth Aug 16 '14 at 10:18
• 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? – user146358 Aug 16 '14 at 10:27

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 $t$.

• 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. – user146358 Aug 16 '14 at 12:57
• 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. – Did Aug 16 '14 at 13:03
• 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. – user146358 Aug 16 '14 at 13:05
• Quote: "the references there do not lead to formal definitions". – Did Aug 17 '14 at 9:20