m-dependent events and the Borel Cantelli lemma

Question

Let $A_1, A_2 , \dots$ be events such that

1. $A_i$ and $A_j$ are independent whenever $|i-j|\geq m$ for some $m>0$.

2. $\sum_{n=1}^{\infty} P(A_n) = \infty$.

Claim. $P(\limsup A_n) = 1$.

I tried to prove this by considering the random variables $N_n = \mathbb{1}_{A_1} + \cdots + \mathbb{1}_{A_n}$ and

$$P(N_n \leq x) \leq \frac{\text{var}(N_n)}{(E(N_n)-x)^2} $$

for each $x$.

However, I failed to prove the claim.

Any help would be appreciated!

Answer 1

There exists some $r\in\left\{0,\dots,m-1\right\}$ such that $\sum_{n=1}^{\infty}\mathbb P\left(A_{nm +r}\right)=\infty$. Otherwise, all the series $\sum_{n=1}^{\infty}\mathbb P\left(A_{nm +r}\right)$, $r\in\left\{0,\dots,m-1\right\}$, would be convergent, and we would reach a contradiction.

Define $B_n:=A_{nm+r}$. By the assumptions, we can use the a version of the Borel-Cantelli lemma for pairwise independent events. To conclude, if infinitely many $B_n$'s are realized, then infinitely many $A_n$'s are realized.