Slow Convergence in Generalized Central Limit Theorems

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 $L^{\mathrm{\infty }}$ of both the densities and distributions of ${\sum }_i^n{X}_i\mathrm{/}(n^{1\mathrm{/}\alpha }L(n))$ 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 $n$.