Theorems like the Lovász Local Lemma?

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.)

• – Sam Hopkins Mar 14 at 17:53
• Talagrand’s concentration inequality in particular is very powerful for this kind of thing. – Sam Hopkins Mar 14 at 18:09

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, 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).