(review question for self-study)
If I have a sequence of densities $f_n$ and a limit density $g$, that is, $f_n \to g$ pointwise, does it follow that the random variables $X_n$ with pdfs $f_n$ converge in law to the random variable $Y$ with pdf $g$? If not, can you provide a counter example?
So I've previously been able to show that $\int |f_n(x) - g(x)|dx$, but only over the whole real line, making use of the fact that $\int f_n(x) - g(x) dx = 0$ and applying DCT.
So it seems that this may be false? But I can't think of a counter example. Clearly the $f_n$ can't be bounded, but I'm having trouble coming up with a counterexample where $f_n$ actually converges pointwise to a density with a misbehaving CDF.