Bounds to the Normal Approximation for Linear Recursions with Two Effects

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.