Bounds to the Normal Approximation for Linear Recursions with Two Effects

TL;DR

This paper extends bounds on the normal approximation for linear recursions to a combined model with two effects, analyzing the sum of two approximately linear processes and providing bounds using Stein's method.

Contribution

It introduces a new analysis for the sum of two approximately linear recursions, extending previous bounds to this more complex model using zero bias transformation.

Findings

Bounds on Wasserstein distance for combined models

Extension of previous results to two-effect linear recursions

Application of Stein's method with zero bias transformation

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.