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.