Slow Convergence in Generalized Central Limit Theorems

Abstract

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha \le 2$. 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 $n$.

To appear in Comptes Rendus de l'Académie des Sciences, Mathématiques

References in corpus (2)

• Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards
• Recurrence properties of a special type of Heavy-Tailed Random Walk

Cited by in corpus (3)

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