# Bounds of the Normal Approximation for Linear Recursions with Two Effects

## Authors

• Mongkhon Tuntapthai

## Keywords:

Hierarchical sequence, Stein's method, Zero bias

## Abstract

Let $X_0$ be a non-constant random variable with finite variance. Given an integer $k\geq&space;2$, define a sequence $\{X_n\}^{\infty}_{n=1}$ of approximately linear recursions with small perturbations $\{\Delta_n\}^{\infty}_{n=0}$ by

$X_{n+1}=\sum_{i=1}^{k}a_{n,i}X_{n,i}+\Delta_{n}&space;\text{&space;for&space;all&space;}&space;n&space;\geq&space;0$

where $X_{n,1},\ldots,X_{n,k}$ are independent copies of the $X_{n}$ and $a_{n,1},\ldots,a_{n,k}$ 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 $C&space;\gamma^{n}$ 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 $n\geq0$, where $\{Y_n\}^{\infty}_{n=1}$ is a sequence of approximately linear recursions with an initial random variable $Y_{0}$ and perturbations $\{\Lambda_n\}^{\infty}_{n=0}$, i.e., for some $\ell\geq2$,

$Y_{n+1}=\sum_{j=1}^{\ell}b_{n,j}Y_{n,j}+\Lambda_{n}&space;\text{&space;for&space;all&space;}&space;n&space;\geq&space;0$

where $Y_{n}$ and $Y_{n,1},\ldots,Y_{n,\ell}$ are independent and identically distributed random variables and $b_{n,1},\ldots,b_{n,\ell}$ are real numbers. Applying the zero bias transformation in the Stein'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.