Uniform convergence of cdf implies uniform convergence of quantile functions

Question

Problem 21.1 of the book 'Asymptotic Statistics' by Aad van der Vaart reads the following

Suppose that $F_n \to F$ uniformly. Does this imply that $F_n^{-1} \to F^{-1}$ uniformly or pointwise? Give a counterexample.

I think uniform convergence of cdf's definitely implies pointwise convergence of quantile functions but this is simply because uniform convergence of cdfs implies pointwise convergence of cdfs which subsequently gives pointwise convergence of quantile functions (Lemma 21.2 of the book).

However, I am not sure whether we can also obtain uniform convergence of quantile functions. From the sentence "Give a counterexample" I think that we can't but I cannot come up with a counterexample.

Answer 1

You do not even get pointwise convergence, only pointwise convergence in continuity points of the limiting quantile function. As a counterexample, define for $n > 2$ $$F_n (x) := \begin{cases} 0 \quad &x < 0 \\ 1/2-1/n \quad &0 < x < 1 \\1 \quad & x\geq 1 \end{cases}$$ With $$F(x) := \begin{cases} 0 \quad &x < 0 \\ 1/2 \quad &0 < x < 1 \\1 \quad & x\geq 1 \end{cases}$$ we then have $F_n \to F$ uniformly. However, the median of $F_n$ is equal to $1$, while the median of $F$ is equal to $0$.