Let $(X_i)_i$ be a sequence of iid random variables, s.t. for some sequences $a_n, b_n$ the normalized sum $$Z_n=\frac{X_1+\dots+X_n}{b_n}-a_n$$ converges weakly to an $\alpha$-stable distributed random variable $Z$ with density $q$. It is known that if $Z_n$ has a bounded density $p_n$, then we have a uniform local limit theorem $$\Delta_n=\sup_{x\in\mathbb{R}}|p_n(x)-q(x)|\to 0.\tag{1}\label{1}$$ Question: What can we say about the rate of convergence $\Delta_n\to 0$?
I found a result concerning $L^p$-convergence, see Banys 1975, but no asymptotic expansion or at least some large/small-o estimates for the case $p=\infty$.
I would also appreciate any comments concerning related topics, e.g. rate of convergence for the corresponding characteristic functions (since \eqref{1} is shown by Fourier-inversion).
Thanks in advance