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

## Keywords:

Hierarchical sequence, Stein's method, Zero bias## Abstract

Let be a non-constant random variable with finite variance. Given an integer , define a sequence of approximately linear recursions with small perturbations by

where are independent copies of the and are real numbers. In 2004, Goldstein obtained bounds on the Wasserstein distance between the standard normal distribution and the law of which is in the form for some constants > and < < .

In this article, we extend the results to the case of two effects by studying a linear model for all , where is a sequence of approximately linear recursions with an initial random variable and perturbations , i.e., for some ,

where and are independent and identically distributed random variables and are real numbers. Applying the zero bias transformation in the Stein's equation, we also obtain the bound for . Adding further conditions that the two models and are independent and that the difference between variance of and is smaller than the sum of variances of their perturbation parts, our result is the same as previous work.