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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.)

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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).

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Does exchangeability qualify as a "relaxed form of independence"? There are a number of results for exchangeable sequences, for example Hong & Lee for a Weak Law of Large Numbers or Fortini, Ladelli & Regazzini for a Central Limit Theorem;

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