Convergence of density functions of random variables.

Question

Various kinds of convergence of random variables are described in wikipedia, but a convergence of density functions is not described. Is it not so interesting?

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.).

Answer 1

The following arguments might explain up to a certain level why convergence of density functions is not very fruitful.

---

It is quite well possible that limit $f$ exists but is not a density function in this situation.

Also if $g$ is a density function then so is every function $g+\mathbf1_A$ where $A$ has zero-measure, and this of the same distribution.

So actually you cannot speak of the density function but at most of a density function.

Okay, in many cases there is a unique one that is continuous and in that sense canonical, but not always.

In short: density functions are (too) capricious.

Answer 2

The question is too broad but we can draw some interesting conclusions from convergence of densities. In particular $\sup_A |P(X_n \in A)-P(X \in A)| \to 0$ which of course implies convegrence in distribution. However we cannot assert convergence in probability or almost sure convergnce.