Convergence in distribution and convergence in Kolmogorov distance Let $X, Y$ be two random variables with laws $F$ and $G$ respectively. The Kolmogorov distance between these two laws is defined as:
$$
d_{Kol}(F, G) = \sup_{x \in \mathbb R} |\mathbb P(X \leq x) - \mathbb P(Y \leq x)|
$$
Now consider a sequence of random variables $X_n \sim F_n$ and a random variable $X \sim F$. It is easy to see that $d_{Kol}(F_n, F) \rightarrow 0$ implies $X_n \rightarrow_d X$ ($\rightarrow_d$ is for convergence in distribution). My questions are:

*

*Under what assumptions, the other direction is also true? (that is, convergence in distribution implies convergence in Kolmogorov distance). This paper (the first couple of paragraphs) claims this to be true when $F$ is continuous, but I got lost when trying to prove it when using the second Dini's theorem.

*In general, can convergence in probability imply convergence in Kolmogorov distance (without assuming continuous target law)?

Any ideas and insights would be really appreciated.

The distinction between convergence in distribution and convergence in Kolmogorov norm is the reason for the importance of the Glivenko-Cantelli theorem on convergence of the empirical cdf $\mathbb{F}_n$ to the true cdf $F$.
Convergence in distribution implies convergence of $F_n$ to $F$ pointwise at points where $F$ is continuous (and uniformly on intervals that don't contain a discontinuity of $F$).  But the Glivenko-Cantelli theorem is stronger, and says the convergence is uniform on the whole domain even in the presence of discontinuities. In contrast to the counterexample with convergence in distribution, where the discontinuity in $F_n$ moves as $n$ increases, the discontinuities in $\mathbb{F}_n$ (the ones that are due to discontinuity of $F$, not just finite sample size) stay in the same place.
