A quantitative discounted central limit theorem using the Fourier metric

Abstract

The discounted central limit theorem concerns the convergence of an infinite discounted sum of i.i.d. random variables to normality as the discount factor approaches $1$. We show that, using the Fourier metric on probability distributions, one can obtain the discounted central limit theorem, as well as a quantitative version of it, in a simple and natural way, and under weak assumptions.