Difference between $\lim P[...]$ and $P[ \lim ]$

Question

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

Answer 1

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 $t$.