TITLE: What is the rate of convergence in the uniform local limit theorem?
QUESTION [6 upvotes]: 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

REPLY [2 votes]: (A late answer since I recently came across a similar question.) 
The best result I found is by Basu and Maejima: 

If the distribution of $X_1$ is absolutely continuous with density $f_{X_1}$ and belongs to the normal domain of attraction of an $\alpha$-stable random variable (strictly stable if $\alpha\in (0,1]$) with density $f_Z$, the characteristic function of $X_1$ is integrable to some power and 
  $$
\int_{-\infty}^\infty x^{\lfloor\alpha\rfloor+1}|f_{X_1}(x) - f_Z(x)|dx<\infty,\tag{1}
$$
  then (with $Z_n$ as above)
  $$
\sup_{x\in \mathbb{R}}(1+|x|^{\alpha})|f_{Z_n}(x) - f_Z(x)|= O(n^{1-(1+\lfloor\alpha\rfloor)/\alpha}),\quad n\to\infty. \tag{2}
$$

The condition (1) about existence of pseudomoment is quite restrictive. Rachev and Yukich pose instead assumptions on some "ideal metrics". Such assumptions are weaker than (1) and are close to necessary, but I found them rather hard to verify.