Bounds of the Normal Approximation for Linear Recursions with Two Effects
Keywords:
Hierarchical sequence, Stein's method, Zero biasAbstract
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.
