{
  "id": "1911.06444",
  "version": "v1",
  "published": "2019-11-15T01:46:30.000Z",
  "updated": "2019-11-15T01:46:30.000Z",
  "title": "Bounds to the Normal Approximation for Linear Recursions with Two Effects",
  "authors": [
    "Mongkhon Tuntapthai"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $X_0$ be a non-constant random variable with finite variance. Given an integer $k\\ge2$, define a sequence $\\{X_n\\}_{n=1}^\\infty$ of approximately linear recursions with small perturbations $\\{\\Delta_n\\}_{n=0}^\\infty$ by $$X_{n+1} = \\sum_{i=1}^k a_{n,i} X_{n,i} + \\Delta_n \\quad \\text{for all } n\\ge0$$ where $X_{n,1},\\dots,X_{n,k}$ are independent copies of the $X_n$ and $a_{n,1},\\dots,a_{n,k}$ are real numbers. In 2004, Goldstein obtained bounds on the Wasserstein distance between the standard normal distribution and the law of $X_n$ which is in the form $C \\gamma^n$ for some constants $C>0$ and $0 < \\gamma < 1$. In this article, we extend the results to the case of two effects by studying a linear model $Z_n=X_n+Y_n$ for all $n\\ge0$, where $\\{Y_n\\}_{n=1}^\\infty$ is a sequence of approximately linear recursions with an initial random variable $Y_0$ and perturbations $\\{\\Lambda_n\\}_{n=0}^\\infty$, i.e., for some $\\ell \\ge2$, $$Y_{n+1} = \\sum_{j=1}^\\ell b_{n,j} Y_{n,j} + \\Lambda_n \\quad \\text{for all } n\\ge0$$ where $Y_n$ and $Y_{n,1},\\dots,Y_{n,\\ell}$ are independent and identically distributed random variables and $b_{n,1},\\dots,b_{n,\\ell}$ are real numbers. Applying the zero bias transformation in the Stein\\rq s equation, we also obtain the bound for $Z_n$. Adding further conditions that the two models $(X_n,\\Delta_n)$ and $(Y_n,\\Lambda_n)$ are independent and that the difference between variance of $X_n$ and $Y_n$ is smaller than the sum of variances of their perturbation parts, our result is the same as previous work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-15T01:46:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "normal approximation",
      "approximately linear recursions",
      "random variable",
      "real numbers",
      "standard normal distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}