I’ve been looking for a quantitative notion of the Borel-Cantelli lemma along the following lines:


Let $(X,\Omega,p)$ be a probability space, and let $(A\_n)\_n\subseteq \Omega$ be a sequence of measurable sets s.t $p(A\_n)\geq 1-\delta$ for all $n$ (for some small $\delta>0$).


Then of course, the Borel-Cantelli lemma tells us that $p(\limsup A\_n)>0$, and in fact a trivial observation from the proof of the lemma shows $p(\limsup A\_n)\geq 1-\delta$.


However, I am interested in something further, relating to the lower density of sequences of indices to which elements belong: is there $\alpha>0$ s.t
$$p\left(\left\{x\in X: \exists n\_k\uparrow \infty\text{ with }\underline{d}((n\_k)\_k)\geq 1-\alpha\text{ s.t }x\in A\_{n\_k},\forall k\geq 0\right\}\right)\geq 1-\alpha?$$
where $\underline{d}(\cdot)$ denotes the lower-density of a sequence. In particular, can $\alpha>0$ be made small when $\delta>0$ is small?


Intuitively it seems to make sense, but so far no success in proving, and also searching online yielded nothing. Any interesting counter examples are also welcome.


Thanks.



Quantitative Borel-Cantelli
Imagine that for all $k=1,2,\ldots$ all events $A\_n$ for $n\in [k!,(k+1)!-1)$ are the same event $C\_k$. Then if $x$ belongs to all $A\_i$ along a subsequence of positive density yields that it belongs to all but finitely many $C\_k$. It may well appear that there is no such $x$.


