Slow Convergence in Generalized Central Limit Theorems

TL;DR

This paper investigates the slow convergence rates in generalized central limit theorems for stable laws, showing that convergence is logarithmically slow unless the scaling function is trivial, impacting the accuracy of asymptotic laws.

Contribution

It establishes that convergence in the non-normal domain of attraction for symmetric alpha-stable laws is at best logarithmic, highlighting limitations of existing asymptotic approximations.

Findings

Convergence rate is logarithmic for non-trivial slowly varying functions.

Asymptotic laws with sqrt(n log n) scaling are accurate only for exponentially large n.

Implication that many physical process models may have limited practical accuracy.

Abstract

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and distributions of $\sum_i^n X_i/(n^{1/\alpha}L(n))$ is at best logarithmic if $L$ is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using central limit theorems with $\sqrt{n\log n}$ scaling and Gaussian limiting distributions. Our result implies that such asymptotic laws are accurate only for exponentially large $n$.