I was reading a paper titled Random Oracles with(out) Programmability. The last paragraph of section 2.3 reads:
[Using our novel approach] there is no need to apply well-known classical asymptotic (and uniform) derandomization techniques based on the Borel-Cantelli lemma. To the best of our knowledge, this approach is novel to this paper.
I took a look at Wikipedia's entry for Borel–Cantelli lemma, and almost grasped the idea. However, I couldn't still figure out how it relates to derandomization. In addition, I don't understand the meaning of "asymptotic" and "uniform" in the aforementioned paragraph.
PS: Googling for Borel-Cantelli and derandomization will show several interesting results, but I don't have enough background to understand them well.