 Poisson approximation bound for dependent Bernoulli variables

For Bernoulli random variables $X_1,\dots ,X_n$, which need not be independent, put $p_i=\mathbb{P}(X_i=1)$, $S={\sum }_i{X}_i$, and $\lambda ={\sum }_i{p}_i$. ThenThe dependence penalty is the total correlation. The proof compares the joint Bernoulli law with the product of Poisson laws of means $p_i$, then applies the data processing inequality for relative entropy to addition.