# Convergence of probability density function

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?

• Could you tell me how to prove the theorem?I want to know the proof of the claim in the first half.If $p_{n_0}(x)$ is bounded for some $n_0$, why $p_n(x)\to\frac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$ uniformly? – Takeo 2 days ago