= Stein-Chen bound with the Poisson Stein factor
{c}
Let $W=\sum_i I_i$ and $\lambda=\sum_i p_i>0$, where each <indicator random variable> $I_i$ is independent of all indicators outside $\{i\}\cup J_i$ jointly. Write $V_i=\sum_{j\in J_i}I_j$ and $W_i=W-I_i-V_i$. For a test-event solution of the <Poisson Stein equation>, independence gives
$$
\lambda\mathbb Eg(W+1)-\mathbb EWg(W)
=\sum_i p_i\mathbb E[g(W+1)-g(W_i+1)]
+\sum_i\mathbb EI_i[g(W_i+1)-g(W)].
$$
Telescoping each difference and using $\|\Delta g\|_\infty\leq\min(1,\lambda^{-1})$ bounds the <total variation distance> by that factor times $\sum_i p_i^2+\sum_i\sum_{j\in J_i}(p_ip_j+\mathbb EI_iI_j)$. Directed dependence neighborhoods are allowed. If $\lambda=0$, the count is identically zero.
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