Bounds of the Normal Approximation for Linear Recursions with Two Effects

Let $X_0$ be a non-constant random variable with finite variance. Given an integer $k \ge 2,$ define a sequence $\{X_n\}^\infty_{n=1}$ of approximately linear recursions with small perturbations $\{\Delta_n\}^\infty_{n=0}$ by \[ X_{n+1} = \sum^k_{i=1} a_{n,i}X_{n,i} + \Delta_n \text{ for all } n \ge 0 \] 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 $C\gamma^n$ for some constants $C > 0$ and $0 < γ < 1$.

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 $n \ge 0$, 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 $\ell \ge 2$, \[ Y_{n+1} = \sum^\ell_{j =1} b_{n,j}Y_{n,j} + \Lambda_n \text{ for all } n \ge 0\] 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.