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, 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).