There are various kinds of ([convergence of random variables](https://en.wikipedia.org/wiki/Convergence_of_random_variables)) but I have never read about convergence of density functions.


Let $X\_1, X\_2, \dots, X$ be random variables $\Omega \to \mathbb{R}$ and $f\_1, f\_2, \dots, f$ be those density functions $\mathbb{R} \to \mathbb{R}$. The convergence of density functions is $\lim\_{n\to\infty}f\_n = f$ (a.e.) or $\lim\_{n\to\infty}\int\_\mathbb{R}|f\_n-f|=0$.


I think it would be interesting, for example, to prove an analog of the central limit theorem not on cumulative distribution functions but on density functions.


Question: Is there a research on convergence of density functions?



Convergence of probability density function
Such theorems, called local (central) limit theorems, are well known; see e.g. [The Encyclopedia of Mathematics](https://www.encyclopediaofmath.org/index.php/Local_limit_theorems). 


