I am learning about local limit theorems. The following example is probably why we don't have a "convergence in density/pmf."

Ex: $X_1,X_2,\ldots$ is a sequence of independent RVs with mean $a$ and variance $\sigma^2$, then the distribution function $F_n(x)$ of the normalized sum $Z_n=\frac{\sum_{i=1}^n (X_i-a)}{\sigma \sqrt n}$ converges to the normal distribution function. However, this does not imply the convergence of density of $p_n(x)$ of $Z_n$ to $\frac{e^{-x^2/2}}{\sqrt{2\pi}}.$

What conditions can be included to ensure convergence of PDFs/PMFs?