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

Zhonggen Su, Xiaolin Wang

Published 2022-09-20Version 1

Let $(X_i, i\in J)$ be a family of locally dependent non-negative integer-valued random variables, and consider the sum $W=\sum_{i\in J}X_i$. We establish general error upper bounds for $d_{TV}(W, M)$ using the three-parametric perturbation form of Stein operator, where the target variable $M$ is either the mixture of Poisson distribution and binomial or negative binomial distribution, and the three parameters are delicately determined by guaranteeing $M$ and $W$ are identical in the first three factorial cumulants. As applications, we discuss in detail the classic local dependence structure examples like $(1, 1)$-runs, ($k_{1},k_{2}$)-runs and $k$-runs, and attain $O(|J|^{-1})$ error bounds under some special cases by exactly figuring out the smoothness of the conditional distributions of $W$ upon partial sums. The obtained result is to some extent a significant improvement of Pek\"{o}z [Bernoulli, 19 (2013)] and Upadhye, et al. [Bernoulli, 23 (2017)].