Quote:
Originally Posted by Private Message
I'm trying to intuitively grasp what's going on in the [Borel-Cantelli] lemma at the moment but I'm having a bit of trouble and I was wondering whether you could provide some help
The Borel-Cantelli lemma states that if
then
In this post, I will try to explain what this means. First of all,
Working from the inside out, we first need to understand what the sup of a collection of sets is.
Subsets of the sample space can be partially ordered by inclusion. The supremum of a collection of elements in a partially ordered set is its least upper bound. So
should be an upper bound for all the sets A
j, j ≥ n. In other words, B
n is a set, and it should satisfy A
j ⊂ B
n for all j ≥ n.
Also, it should be the
least upper bound. So it should be the smallest set with this property. From here, it is not hard to work out that
So we have determined that
Next, what does it mean to take a limit of a sequence of sets? Well, in general, this may not be a well-defined concept without further specification. But this is a special case, because the sequence B
n happens to be a monotone decreasing sequence. That is,
That means that the limit is just the greatest lower bound (aka, the infimum). In exactly the same way as above, we can work out that the infimum is the intersection of all the B
n's. So in the end, we have
Finally, I will leave it as an exercise to prove that
In other words, this is the event that infinitely many of the A
j's occur. Another way to say this is that the A
j's happen "infinitely often". In probability, we abbreviate this with "i.o.", so that
In words, then, the Borel-Cantelli lemma says that if the probabilities of the A
j's are summable, then the probability that they happen infinitely often is 0.
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Incidentally, a collection of declarative sentences can be partially ordered by logical implication. In that case, the supremum of two sentences, A and B, is the sentence "A or B"; and the infimum is the sentence "A and B". This is exactly why, in probability, union means "or" and intersection means "and". If, in this context, {A
n} is a sequence of such sentences, then the limsup of this sequence is the assertion that infinitely many of the sentences A
1, A
2, ... are true.