Let $(F_n)_{n\ge 1}$ be a sequence of distribution functions on $\mathbb R$, and let $F:\mathbb R\to[0,1]$ be a distribution function. Thus each $F_n$ and $F$ is nondecreasing, right-continuous, satisfies $\lim_{x\to-\infty}F_n(x)=\lim_{x\to-\infty}F(x)=0$, and satisfies $\lim_{x\to\infty}F_n(x)=\lim_{x\to\infty}F(x)=1$. Assume that for every continuity point $x\in\mathbb R$ of $F$,

\begin{align*} \lim_{n\to\infty}F_n(x)=F(x). \end{align*}

For each $n\ge1$, define the [quantile function](/page/Quantile%20Function) $Q_n:(0,1)\to\mathbb R$ by

\begin{align*} Q_n(p):=F_n^{-1}(p):=\inf\{x\in\mathbb R:F_n(x)\ge p\}, \end{align*}

and define $Q:(0,1)\to\mathbb R$ by

\begin{align*} Q(p):=F^{-1}(p):=\inf\{x\in\mathbb R:F(x)\ge p\}. \end{align*}

If $p\in(0,1)$ is a continuity point of $Q$, then

\begin{align*} \lim_{n\to\infty}Q_n(p)=Q(p). \end{align*}