Poisson approximation with dependency neighborhoods ($d_{TV}(\mathcal{L}(W),\mathrm{Poisson}(\mu ))\le b_1+b_2$)

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 $\mu ={\sum }_a{p}_a$, $b_1={\sum }_a{\sum }_{b\in B_a}{p}_a{p}_b$ and $b_2={\sum }_a{\sum }_{b\in B_a\setminus \{a\}}\mathbb{E}(I_a{I}_b)$. The Poisson Stein equation and its bound $\Vert \Delta f{\Vert }_{\infty }\le 1$ give the displayed total variation distance estimate. Stronger Stein factors can improve the bound when $\mu$ is large.