The Lovász Local Lemma gives a probability bound in a context where there are many events that are "not quite" independent.


What other theorems exist in this genre? That is, what other theorems have a hypothesis of the form "Let events E\_1, E\_2, ... satisfy [relaxed form of independence]" and a conclusion of the form "Then the probability of [compound event] satisfies [inequality]"?


(I hope this question isn't too broad. I frequently encounter problems with events that are "almost independent", either in the sense that most subsets are independent or in the sense that the probabilities of compound events are well-approximated by assuming independence, and I am looking for general tools that may be helpful when these situations come up.)



Theorems like the Lovász Local Lemma?
A large number of results for sums $W$ of possibly dependent indicators of events (that is, for sums of possibly dependent Bernoulli random variables) $X\_i$ have been obtained by the Chen--Stein method. See e.g. [Theorem 1](https://projecteuclid.org/euclid.ss/1177012015), which gives an upper bound on the total variation distance between the distribution of such a sum $W$ and a corresponding Poisson distribution in terms of certain characteristics $b\_1,b\_2,b\_3$ of the strength of the dependence between the $X\_i$'s (defined in formulas (4)--(6) of that paper). 


