Counterexamples concerning the central limit theorem The central limit theorem states that, if $X$ is a random variable with finite variance $\sigma^2$ and expected value $\mu$, and if $(X_n)$ is a sequence of independent random variables identically distributed like $ X $, then
\begin{equation}
Z_n = {\frac{{\overline{X}}_n-\mu}{\sqrt{\sigma^2/n}}}\ \rightsquigarrow N(0,1),
\end{equation}
where ${\overline{X}}_{n} = \frac{1}{n} \sum_{i=1}^{n} X_i$ and $\rightsquigarrow$ means convergence in distribution, which in this case is equivalent to the pointwise convergence of the cdf of $Z_n$ to the cdf of a $N(0,1)$.
Suppose moreover that $X_n$ is absolutely continuous for all $n$‘s. Then $Z_n$ is absolutely continuous for all $n$‘s. Are there examples of such sequences $(X_n)$, such that the pdf of $Z_n$ does not converge pointwise to the pdf of a $N(0,1)$, not even almost everywhere (w.r.t. the Lebesgue measure on $\mathbb R$)?

I can recommend
A Few Counter Examples Useful in Teaching Central Limit Theorems., www.jstor.org/stable/24590348.
As well as chapter 17 of the great book Counterexamples in Probability by Jordan Stoyanov.
