Probability Theory – Dependence and Second Borel-Cantelli Lemma

Question

I'll put the problem and then I'll explain my problem.

Problem: Let ${A_n}$ be events such as

$\operatorname{Cov}(I_{A_i},I_{A_j})=E[I_{A_i}I_{A_j}]-E[I_{A_i}]E[I_{A_j}]\leq 0,\ \forall i\neq j\tag{1}$

If $\sum P(A_i)=\infty$ then $P[\lim \sup A_n]=1$.

Answer: By (1) we have that $\operatorname{Cov}(I_{A^c_i},I_{A^c_j})\leq 0$ too.
So $P(\lim \inf A_n^c)=P(\lim \bigcap A_n^c)=\lim P(\bigcap A_n^c)\overset{Q!} {\leq} \lim \prod P(A_n^c)= \lim \prod (1-P(A_n))\leq \lim e^{-\sum P(A_n)}=0$

The detail is that in the inequality marked with a "Q!" I used that $P(\bigcap A_n)\leq \prod P(A_n)$. It is intuitive but i couldn't prove through the problem statement. But it is interesting result. We could use B-C lemma even for correlated events, since they are negatively correlated. What do you guys think about it?

Answer 1

I think Fristedt & Gray's book on probability proves a version of Borel--Cantelli that assumes only nonpositive correlation rather than independence. In particular, that means pairwise independence is enough.

Later edit: Here's what I find in A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray, page 79:

Lemma 5. [Borel-Cantelli] Let $(A_1,A_2,\ldots)$ be a sequence of events in a probability space $(\Omega,\mathcal{F},P)$. Assume that for each $i\ne j$, the events $A_i$ and $A_j$ are either negatively correlated or uncorrelated. Let $A=\lim\sup_{n\to\infty} A_n$. If $\sum_{n=1}^\infty P(A_n)=\infty,$ then $P(A)=1$.