Approximation of sums of locally dependent random variables via perturbations of Stein operator

Let $(X_i,\, i\in J)$ be a family of locally dependent nonnegative integer-valued random variables (r.v.'s), and consider a sum $W=\sum_{i\in J}X_i$. Applying the Stein method, we prove an upper bound for total variation distance $d_{\mathrm{TV}}(W, M)$, where an approximating r.v. $M$ has the distribution, which is a mixture of Poisson distribution with either binomial or negative binomial distribution. As a corollary of general results, we get approximation errors of order $O(|J|^{-1})$ for the distributions of ($k_1,k_2$)-runs and $k$-runs. The results obtained are significantly better than the previously known estimates, e.g., $O(|J|^{-0.5})$ [E. Peköz, A. Röllin, and N. Ross, Bernoulli, 19 (2013), pp. 610–632] and $O(1)$ [N. Upadhye, V. Čekanavičius, and P. Vellaisamy, Bernoulli, 23 (2017), pp. 2828–2859].