Refining the Central Limit Theorem Approximation via Extreme Value Theory

TL;DR

This paper introduces a novel approximation method for sums of i.i.d. random variables with Pareto-like tails, combining extreme value theory and normal approximation to improve accuracy over traditional methods, especially for heavy-tailed distributions.

Contribution

The authors propose a new hybrid approximation technique that improves the accuracy of the Central Limit Theorem for heavy-tailed distributions by integrating extreme value theory.

Findings

Significantly reduced error rates compared to normal approximation.

Effective for distributions with finite variance and less than three moments.

Applicable to some infinite variance distributions.

Abstract

We suggest approximating the distribution of the sum of independent and identically distributed random variables with a Pareto-like tail by combining extreme value approximations for the largest summands with a normal approximation for the sum of the smaller summands. If the tail is well approximated by a Pareto density, then this new approximation has substantially smaller error rates compared to the usual normal approximation for underlying distributions with finite variance and less than three moments. It can also provide an accurate approximation for some infinite variance distributions.