probability - Hypotheses on ${X_n}_{n=1}^{infty}$ and $X$ so that $lim_n f_{X_n}(x)=f_X(x)$ for a.e. $xinmathbb{R}$.

Let $\{X_n\}_{n=1}^\infty$ and $X$ be absolutely continuous random variables on a probability space $(\Omega,\mathcal{F},P)$, with density functions $f_{X_n}(x)$ and $f_X(x)$. Are there any hypotheses on $\{X_n\}_{n=1}^\infty$ and/or $X$ that guarantee the convergence $\lim_n f_{X_n}(x)=f_X(x)$ for a.e. $x\in\mathbb{R}$?

The most related result that I know is the following: if $\{X_n\}_{n=1}^\infty$ does not converge in law to $X$, then it not possible to have $\lim_n f_{X_n}(x)=f_X(x)$ for a.e. $x\in\mathbb{R}$, by Scheffé's Lemma. But this result does not give a sufficient condition for $\lim_n f_{X_n}(x)=f_X(x)$.