{
  "id": "2209.09770",
  "version": "v1",
  "published": "2022-09-20T14:54:58.000Z",
  "updated": "2022-09-20T14:54:58.000Z",
  "title": "Approximation of Sums of Locally Dependent Random Variables via Perturbation of Stein Operator",
  "authors": [
    "Zhonggen Su",
    "Xiaolin Wang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "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)].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-20T14:54:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "locally dependent random variables",
      "stein operator",
      "general error upper bounds",
      "local dependence structure examples",
      "non-negative integer-valued random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}