Convergence in distribution to a uniform distribution

Question

Say $(X_n)_{n\geq 1}$ is a sequence of continuous random variables, and let $F_{X_n}$ be their cumulative distribution functions respectively.

Is is true that, for every $0 \leq x \leq 1$, $$ \lim_{n \longrightarrow \infty} F_{X_n}\left(F^{-1}_{X_n}(x)+1/n\right)=x?$$

Thanks!

Answer 1

No, e.g., let $F_{X_n}(x)$ be $0$ for $x\le -1/n$, $1$ for $x>0,$ and linear in between. Then for all $n,$ $F^{-1}_{X_n}(1/2)=-1/(2n),$ $F^{-1}_{X_n}(x)+1/n=1/(2n),$ and $F_{X_n}\left(F^{-1}_{X_n}(x)+1/n\right)=1.$ You probably want some Lipshitz condition on the $F_{X_n}$.