Corrections to the Central Limit Theorem for Heavy-Tailed Probability Densities

TL;DR

This paper extends the classical Edgeworth expansions for the CLT to heavy-tailed distributions with diverging moments, providing explicit correction terms that account for asymptotic behaviors beyond the standard approximation.

Contribution

It introduces new correction terms for the CLT applicable to regularly varying distributions with diverging moments, expressed in closed form using special functions.

Findings

Correction terms depend on the parity of available moments.

Logarithmic corrections are necessary for even moments.

Correction terms dominate outside the central region and connect with large deviation asymptotics.

Abstract

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson's integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.