So for a while I have been having a hard time understanding “limit superior” and “limit inferior” of a sequence, but finally I have a grasp on the idea (I hope!), so here goes:
Suppose I have a sequence of events:
A = {A1, A2, …}
this sequence goes to infinity.
there are no special conditions these events have to follow, for example I can have:
A1 = [1, 3, 4]
A2 = [1, 2]
A3 = [11112]
A4 = [12, 11]
A5 = [1, 3, 4, 7]
…
if I decide the sample space to be positive integers.
now if I take the union of the events starting at i = 5:
U{i = (5, inf)}( Ai ) = U{A5, A6, A7, …}
this union is bigger than if I take the union of the events starting at i = 6:
U{i = (6, inf)}( Ai ) = U{A6, A7, A8, …}
Why?
Because the two unions completely overlap except when i = 5, I am taking an additional A5 into the union. Therefore:
U{i = (n, inf)} (Ai)
is decreasing in n. As n gets larger, the union gets smaller.
Now if I take the intersection of the events starting at i = 5:
I{i = (5, inf)}(Ai) = I{A5, A6, A7, …}
this intersection is bigger than if I take the intersection of the events starting at i = 6:
I{i = (6, inf)}(Ai) = I{A6, A7, A8, …}
because I excluded A5 from the intersection.
I{i = (n, inf)} (Ai)
is increasing in n. As n gets larger, the intersection gets larger.
All of that are interesting facts…
Now we get to the main topic:
Define limit superior to be:
limit superior of A = lim{n -> inf} U{i = (n, inf)} (Ai)
Limit superior of A is an event that contains elements that exist in the union of Ai’s as we take n to infinity.
What kind of elements belong in limit superior of A?
Suppose the sample space again is positive integers, and the integer 1 belongs to every event from A1 to A100, then 1 will not be in the limit superior of A, since the union of the events U{i = (101, inf)} (Ai) = (A101, A102, …) will not contain 1, therefore as we take n to infinity 1 will not be in the final resulting event.
So, in order for an element to be in the limit superior of A, it needs to be in infinitely many Ai’s. So that no matter how large we take n to be, one of the Ai’s where i > n will contain this element.
Incidentally limit superior is equivalent to:
I{n = (1, inf)}(U{i = {n, inf}) (Ai))
the intersection of unions of Ai’s for every n.
Now onto limit inferior.
Define limit inferior to be:
limit inferior of A = lim{n->inf} I{i = (n, inf)} (Ai)
Limit inferior of A is an event that contains elements that exist in the intersection of Ai’s as we take n to infinity.
Again what kind of elements belong in the limit inferior?
Obviously, this element has to exist in infinitely many Ai’s, since say if this element only exist in the first 100 As, none of the intersections of Ai’s will contain this element!
BUT! just being in infinitely many Ai’s is not enough for this element to be in the limit inferior!
If you think about the following situation:
Let element 1 belong only to odd number indexed Ai’s (1, 3, 5, …).
Well, 1 definitely exists in infinitely many Ai’s. But 1 does not belong to the limit inferior of A. Because every Intersection of Ai’s contain some even numbered Ai’s, so every resulting intersection will not contain 1!
What additional requirements do you need then?
In order for 1 to be in the limit inferior of A, it needs to not exist in only finitely many Ai’s.
so say if 1 does not exist in the first 100 Ai’s, but exist in every single Ai after A100, then 1 belongs to limit inferior!
Also, limit inferior is equivalent to:
U{n = (1, inf)}(I{i = {n, inf}) (Ai))
the union of intersections of Ai’s for every n.
Conclusion:
For an element in the sample space (say 1!), the following situations can happen:
1. 1 exists in infinitely many Ai’s and 1 does not exist in infinitely many Ai’s.
2. 1 exists in infinitely many Ai’s and 1 does not exist in finitely many Ai’s.
3. 1 exists in finitely many Ai’s and 1 does not exist in infinitely many Ai’s.
1, 2 belongs in limit superior.
Only 2 belongs in limit inferior.
So limit superior contains limit inferior.
Wait, what about 1 exists in finitely many Ai’s and 1 does not exist in finitely many Ai’s?
That’s not possible, since the sequence is infinite XD.
