Let $X,X_1,X_2,\dots$ be continuous random variables in a probability space, whose probability density functions are $f,f_1,f_2,\dots$ respectively. If $f_n\rightarrow f$ almost everywhere (relative to Lebesgue measure,) then $X_n$ converges to $X$ in distribution: $X_n\xrightarrow[]{D} X$