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?



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



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


