Stein-Chen bound with the Poisson Stein factor

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 givesTelescoping each difference and using $\Vert \Delta g{\Vert }_{\infty }\le \min (1,{\lambda }^{-1})$ bounds the total variation distance by that factor times ${\sum }_i{p}_i^2+{\sum }_i{\sum }_{j\in J_i}(p_i{p}_j+\mathbb{E}{I}_i{I}_j)$. Directed dependence neighborhoods are allowed. If $\lambda =0$, the count is identically zero.