The second Borel-Cantelli lemma and its generalizations

The second Borel-Cantelli lemma:

Let $(\Omega, \mathcal{F}, P)$ is a probability space. If the events $A_n$ are independent then $\sum P(a_n)=\infty$ implies $P(A_n i.o.)=1.$

 

The lemma has the following generalizations:

1. If $A_1, A_2, \cdots$ are pairwise independent and $\sum P(a_n)=\infty$ then as $n\rightarrow \infty$

$$\frac{\sum_{m=1}^n I_{A_m}}{\sum_{m=1}^nP(A_m)}\rightarrow 1  a.s.$$

For the proofs of the above results, see Durrett's book.

2. Let $\{A_n\}$ be a sequence of events such that $\sum P(a_n)=\infty$ and

$$\liminf\limits_{n\rightarrow\infty}\frac{\sum_{k=1}^n\sum_{\ell=1}^nP(A_k)P(A_\ell)}{\left(\sum_{k=1}^nP(A_k)\right)^2}=1,$$

then $P(A_n i.o.)=1.$

For proof, see On Cantor's series with convergent $\sum 1/q_n$ by P. Erdos and A. Renyi.