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The second Borel-Cantelli lemma refers to a sequence of independent events $A_n$ such that $\sum_{n=1}^\infty \Pr(A_n)=\infty,$ and says that in this case infinitely many $A_n$ events occur almost surely.

$$\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i=\{A_n \text{ i.o.}\}$$

The indexing in the union considers the union of only events beyond a certain event $n$ in the sequence. So it would comprise the occurrence of any events after $n.$

But what is the intersection? Does it simply mean that all these composite events after every single $n$ do occur?

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  • $\begingroup$ The notation is definitely lacking, since the RHS has no reference for what $n$ is. $\endgroup$
    – abiessu
    Feb 8 at 17:08
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    $\begingroup$ The left hand side simplifies to $\{x : x \in A_n \text{ for infinitely many } n \in \Bbb{N}\}$, so I'm guessing that's what is trying to be conveyed here. $\endgroup$ Feb 8 at 17:11
  • $\begingroup$ You could also note that this event is actually the limit superior of the events, which also intuitively matches the "infinitely often". $\endgroup$ Feb 8 at 19:22
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Just read the intersection as a "for all", and the union as a "there exists". The event $\bigcap_{n=1}^\infty \bigcup_{i=n}^\infty A_i$ then means : "for all $n \geq 1$, there exists $i \geq n$ such that the event $A_i$ occurs". In other words, this means indeed that infinitely many events $A_i$ occur.

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