  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
$n$.
