= Poisson approximation with dependency neighborhoods
{c}
{title2=$d_{TV}(\mathcal L(W),\operatorname{Poisson}(\mu))\le b_1+b_2$}
For indicators $I_a$ with means $p_a$, let $B_a$ contain $a$ and suppose $I_a$ is independent of the entire family outside $B_a$. Put $\mu=\sum_ap_a$, $b_1=\sum_a\sum_{b\in B_a}p_ap_b$ and $b_2=\sum_a\sum_{b\in B_a\setminus\{a\}}\mathbb E(I_aI_b)$. The <Poisson Stein equation> and its bound $\|\Delta f\|_\infty\le1$ give the displayed <total variation distance> estimate. Stronger Stein factors can improve the bound when $\mu$ is large.
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