There are various kinds of (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?