I am interested in what circumstances various zero-one laws in probability theory can be relaxed. In particular, independence is a very important factor in such laws. 


1) Borel-Cantelli Lemma: Let $A\_1, A\_2, \cdots$ be a sequence of events. Then $\mathbb{P}(\limsup\_{n \rightarrow \infty} A\_n) = 0$ if $\displaystyle \sum\_{n = 1}^\infty \mathbb{P}(A\_n) < \infty$, and $\mathbb{P}(\limsup A\_n) = 1$ if $A\_1, A\_2, \cdots$ are pairwise independent and $\displaystyle \sum\_{n = 1}^\infty \mathbb{P}(A\_n) = \infty$.


2) Kolmogorov's zero-one law: If $X\_1, X\_2, \cdots$ are a sequence of random variables, define $H\_n = \sigma(X\_{n+1}, X\_{n+2}, \cdots)$ to be the smallest sigma algebra for which $X\_{n+1}, X\_{n+2}, \cdots$ are measureable. Then it is clear that $H\_n \supset H\_{n+1} \supset \cdots$ Let $H\_\infty = \bigcap\_{n} H\_n$. Now suppose that $X\_1, X\_2, \cdots$ are independent. Then all events $A \in H\_\infty$ satisfy $\mathbb{P}(A) = 0$ or $\mathbb{P}(A) = 1$. 


I am particularly interested in cases where independence, which is a rather strong assumption and difficult to verify, can be replaced by estimates on various moments of the random variables, their correlation, etc. For example, the original statement of the Borel-Cantelli Lemma assumed that the sequence of events are independent, but this has since been weakned to pairwise independence. Any help would be greatly appreciated.



To what extent can the following zero-one laws be relaxed?
Re Borel-Cantelli lemma, if one assumes only the divergence of the series and that $P(A\_n\cap A\_k)\le cP(A\_n)P(A\_k)$ for every distinct $n$ and $k$ large enough, one gets that $P(\limsup A\_n)\ge1/c$. Proof and situation of the problem by V. V. Petrov [here](http://www.math.chalmers.se/Math/Research/Preprints/2001/38.pdf).


Re Kolmogorov's zero-one law, I wonder what kind of weakened *conclusion* you have in mind.


