Let $X_1,X_2,\cdots$ and $Y$ be random variables on $[0,1]$ with smooth density functions $f_1,f_2\cdots$ and $f$. Suppose $X_n\to Y$ in probability. Can we get some convergence of the density functions, at least in $L_1$，like $\lim_{n\to\infty}\int_{[0,1]}|f_n(x)-f(x)|dx=0$?
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$\begingroup$ Yes, if they are all dominated by an integrable function. This follows from the fact that a sequence of random variables converges in probability if and only if every subsequence has a further subsequence that converges almost surely to the same random variable. $\endgroup$– Michael GreineckerCommented Nov 24, 2023 at 7:22
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No. Let $Y$ be uniform on $[0,1]$ and $X_n$ have density $f_n=1+\sin (2\pi nx)$. Then $X_n\to Y$ in distribution. You can represent them on the same probability space $(0,1)$ (with Lebesgue measure) by $X_n(\omega)=F^{-1}_{X_n}(\omega)$ and $Y(\omega)=F^{-1}_{Y}(\omega)$, where $F$ are cumulative distribution functions, so that they in fact converge almost surely. But $\int|f_n(x)-f(x)|$ does not converge to $0$.
