Borel-Cantelli Lemmata to Kochen-Stone Theorem
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Lemmata to Kochen-Stone Theorem
Lemma $10$
Let $A_n$ be a sequence of events with $\ds \sum \map \Pr {A_n} = \infty$ and:
- $\ds \liminf_{k \mathop \to \infty} \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} < \infty$
Then there is a positive probability that $A_n$ occur infinitely often.