Approximation of Sums of Locally Dependent Random Variables via Perturbation of Stein Operator

Let $(X_i,i\in J)$ be a family of locally dependent nonnegative integer-valued random variables, and consider the sum $W={\sum }_{i\in J}{X}_i$. We first establish a general error upper bound for $d_{TV}(W,M)$ using Stein’s method, where the target variable $M$ is either the mixture of Poisson distribution and binomial or negative binomial distribution. As applications, we attain $O(\mathrm{\mid }J{\mathrm{\mid }}^{-1})$ error bounds for ($k_1,k_2$)-runs and $k$-runs under some special cases. Our results are significant improvements of the existing results in literature, say $O(\mathrm{\mid }J{\mathrm{\mid }}^{-0.5})$ in Peköz [Bernoulli, 19 (2013)] and $O(1)$ in Upadhye, et al. [Bernoulli, 23 (2017)].