Is there a notion of Convergence in PDF/PMF 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?
 A: You should look at the old book 

Gnedenko and Kolmogorov: Limit Distributions for Sums of Independent Random Variables. 

where they have examples of  local limit theorems. I assume  many more things were discovered  in the meantime.  Here is an example  of theorem that answers your question.
Theorem.  Suppose that  $(X_n)_{N\geq 1}$ is a sequence of i.i.d.  random variables with common  density (PDF)   $p(x)$. Denote by $p_n(x)$ the  density of $Z_n=X_1+\cdots +X_n$. Assume the  following conditions.


*

*The random variables $X_n$ are $L^2$, i.e.,  $$ \sigma^2:=\int_{-\infty}^\infty x^2 p(x) dx<\infty. $$ 

*There exists $r\in (1,2]$ and a positive  integer $n$ such that $\newcommand{\bR}{\mathbb{R}}$ $p_n\in L^r(\bR)$. (This is guaranteed if e.g.  $p(x)$ is Lipschitz.)


Then
$$
\sigma\sqrt{n}p_n(\sigma\sqrt{n}\; x) \to \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} $$
uniformly with respect to $x\in\bR$.  For a proof and more details see sections 46,47 of the above book.
A: I don't know if this is an answer to your question: Years before I presented the following result in a lecture about weak convergence:
Theorem: Assume that $X_n, n \in N$ are real r.v. with unimodal density $\varphi_n$, $P_0$ a probability measure with unimodal, continuous density $\varphi$, unique maximum value $t^*$ and d.f. $\Phi$, Then if $X_n \to P_0$ in distribution, Then
$\liminf_{n \to \infty} \inf_{t \in R} (\varphi_n(t) - \varphi(t)) = 0$
$\lim_{n \to \infty} \sup_{|t-t^*| \geq \delta} |\varphi_n(t) - \varphi(t))| = 0$ for any $\delta > 0$
and
$\lim_{n \to \infty} \sup_{t \in R} |\varphi_n(t) - \varphi(t))| = 0$
if any $\varphi_n$ is logarithmically concave.
I don't know where you can find this reasult, but the proof is not too hard. There also is a version for discrete r.v.
The usual simple local theorems are special cases.
