1

I am new to probability and I'm trying to understand the concept of $\liminf$ and $\limsup$ of events through an example. I am very comfortable with the technical definitions given by set operations and the interpretations that the $\limsup$ of a sequence of events is the event whose outcomes occur infinitely often in the sequence and the $\liminf$ is the event whose outcomes always occur after some finite time. All of this is well explained in other questions (such as this one: Intuitive interpretation of limsup and liminf of sequences of sets?).

What I would like to do is better understand this concept through an example. The simplest one I can think of is to consider a fair 6-sided die. At time $n$ we roll it and record the outcome, this is a single event $A_n$. Thus we have a sequence of events $\{A_n\}$ and I would like to determine its $\limsup$ and $\liminf$.

I think both the $\limsup$ and $\liminf$ in this example should be $\{1, 2, 3, 4, 5, 6\}$ since all will occur infinitely often and will always occur. Is this right?

Other examples are also welcomed as I'm having trouble coming up with one where $\liminf \neq \limsup$.

CBBAM
  • 5,883
  • 2
  • 6
  • 18
  • $A_n$ is a random variable and ${A_n}$ is a set containing a random variable, not an event. ${A_9 = 4}$ would be an example of notation conventionally used to denote an event – Matthew Towers Dec 21 '23 at 19:05
  • @MatthewTowers Yes I think you're right and maybe this is why I am confused. So the sequence ${A_n}$ is an outcome that has already been realized. Is that right? – CBBAM Dec 21 '23 at 19:09
  • I don't know what that sequence is - it's legitimate to talk about limsup of a sequence of (real valued) random variables but it's not the same thing as limsup of a sequence of sets (e.g. events, which are subsets of the sample space) – Matthew Towers Dec 21 '23 at 19:11
  • @MatthewTowers Maybe it's better if I first look at an example of a sequence of events. – CBBAM Dec 21 '23 at 19:13
  • OK, but you haven't defined any sequences of events. Any event in your example is a subset of ${1,2,3,4,5,6}$. $A_n$ isn't an event, it's a random variable. – Matthew Towers Dec 21 '23 at 19:17
  • @MatthewTowers So if we were to carry out the experiment and got the sequence ${1, 2, 2, 6, 1, 5, 3, 3, 3, 4, \ldots}$ would this be an event or instead a sequence of events? – CBBAM Dec 21 '23 at 19:20

1 Answers1

1

Lets roll the fair dice infinitely many times. The probability that 6 will appear infinitely many times is 1 (The example sequence: 1, 6, 1, 6, 2, 6, 3, 3, 3, 6,...), while the probability of it appearing every single time from some point in time to infinity is 0 (The example sequence: 1, 2, 2, 5, 4, 6, 6, 6, 6, 6,...). You may also take a look at Borel-Canteli theorems.

Vuk Jovovic
  • 748
  • 1
  • 9
  • Thank you, I think this solves most of my confusion! So the $\liminf$ is the empty set but how would we write the $\limsup$? I think it should be something like ${1, 2, 3, 4, 5, 6}$, but wouldn't this correspond to the event where we get $1-6$? So maybe the $\limsup$ should be ${{1}, {2}, {3}, {4}, {5}, {6}}$? – CBBAM Dec 21 '23 at 19:11
  • 1
    Actually, you need to think in terms of entire sequences here. – Vuk Jovovic Dec 21 '23 at 19:59
  • Thank you, I understand now. – CBBAM Dec 21 '23 at 20:40