Theorem: Borel-Cantelli Lemma 10 to Kochen-Stone Theorem

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.

Proof:
Fix $\ell < k$.
Let $\ds X = \sum_{n \mathop = \ell}^k 1_{A_n}$.
It follows that:
:$\ds \expect X = \sum_{n \mathop = \ell}^k \Pr(A_n)$
and:
:$\ds \expect {X^2} = \sum_{\ell \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m}$
Using the Paley-Zygmund Inequality for $\theta = 0$, we obtain:
{{begin-eqn}}
{{eqn | l = \map \Pr {\bigcup_{n \mathop = \ell}^k A_n}
      | r = \map \Pr {X > 0}
      | c = 
}}
{{eqn | o = \ge
      | r = \frac {\ds \paren {\sum_{n \mathop = \ell}^k \map \Pr {A_n} }^2} {\ds \sum_{\ell \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} }
      | c = 
}}
{{eqn | o = \ge
      | r = \frac {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} - \sum_{n \mathop = 1}^{\ell - 1} \map \Pr {A_n} }^2} {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} - \sum_{1 \mathop \le m, n \mathop < \ell} \map \Pr {A_n \cap A_m} }
      | c = 
}}
{{end-eqn}}
Now, it holds that:
:$\ds \map \Pr {A_n \text { occurs i.o.} } = \lim_{\ell \mathop \to \infty} \lim_{k \mathop \to \infty} \map \Pr {\bigcup_{n \mathop = \ell}^k A_n}$
We have:
:$\ds \frac {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} - \sum_{n \mathop = 1}^{\ell - 1} \map \Pr {A_n} }^2} {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} - \sum_{1 \mathop \le m, n \mathop < \ell} \map \Pr {A_n \cap A_m} } \ge \frac {\ds \left(\sum_{n \mathop = 1}^k \map \Pr {A_n} - \sum_{n \mathop = 1}^{\ell - 1} \Pr (A_n)\right)^2} {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} }$
Fix $l \in \N_1$.
Since $\ds \lim_{k \mathop \to \infty}\sum_{n \mathop = 1}^k \map \Pr {A_n} = \infty$, by assumption, if $k$ is sufficiently large:
:$\ds \paren {\sum_{n \mathop = 1}^ k \map \Pr {A_n} - \sum_{n \mathop = 1}^{\ell - 1} \map \Pr {A_n} }^2 \approx \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2$
So if $k$ is sufficiently large:
:$\ds \frac {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} - \sum_{n \mathop = 1}^{\ell - 1} \map \Pr {A_n} }^2} {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } \approx \frac {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } = \frac 1 {\frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} }$
Suppose:
:$\ds \liminf_{k \mathop \to \infty} \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} = c < \infty$
This means that no matter how large $k$ is, there is always some $k' \ge k$, such that:
:$\ds \frac {\ds \sum_{1 \mathop \le m, n \mathop \le k'} \map \Pr {A_n \cap A_m} } {\ds \paren {\sum_{n \mathop = 1}^{k'} \map \Pr {A_n} }^2} \approx c$
So for infinitely many $k$s:
:$\ds \map \Pr {\bigcup_{n \mathop = \ell}^k A_n} \ge \frac 1 {\frac {\ds \sum_{1 \mathop \le m, n \mathop \le k} \map \Pr {A_n \cap A_m} } {\paren {\ds \sum_{n \mathop = 1}^k \map \Pr {A_n} }^2} } \approx \frac 1 c$
Since $\ds \map \Pr {\bigcup_{n \mathop = \ell}^k A_n}$ is decreasing in $k$, we have, approximately:
:$\ds \lim_{k \mathop \to \infty} \map \Pr {\bigcup_{n \mathop = \ell}^k A_n} \ge \frac 1 c$
Since $l$ was arbitrary:
:$\ds \lim_{l \mathop \to \infty} \lim_{k \mathop \to \infty} \map \Pr {\bigcup_{n \mathop = \ell}^k A_n} \ge \frac 1 c > 0$
{{qed}}
{{Namedfor|Émile Borel|name2 = Francesco Paolo Cantelli|cat = Borel, Émile|cat2 = Cantelli}}
Category:Kochen-Stone Theorem
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