A quantitative discounted central limit theorem using the Fourier metric

TL;DR

This paper presents a simple, natural approach using the Fourier metric to establish and quantify the convergence of discounted sums of i.i.d. variables to a normal distribution as the discount factor nears one.

Contribution

It introduces a new method leveraging the Fourier metric to prove and quantify the discounted central limit theorem under weak assumptions.

Findings

Established the discounted CLT using Fourier metric

Provided a quantitative version of the theorem

Simplified the proof under weak conditions

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.