  In this paper, we revisit the concentration inequalities for the supremum of
the cumulative distribution function (CDF) of a real-valued continuous
distribution as established by Dvoretzky, Kiefer, Wolfowitz and revisited later
by Massart in two seminal papers. We focus on the concentration of the
\emph{local} supremum over a sub-interval, rather than on the full domain. That
is, denoting $U$ the CDF of the uniform distribution over $[0,1]$ and $U_n$ its
empirical version built from $n$ samples, we study
$P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>\epsilon)$ for different
values of $\underline{u},\overline{u}\in[0,1]$. Such local controls naturally
appear for instance when studying estimation error of spectral risk-measures
(such as the conditional value at risk), where $[\underline{u},\overline{u}]$
is typically $[0,\alpha]$ or $[1-\alpha,1]$ for a risk level $\alpha$, after
reshaping the CDF $F$ of the considered distribution into $U$ by the general
inverse transform $F^{-1}$. Extending a proof technique from Smirnov, we
provide exact expressions of the local quantities
$P(\sup_{u\in[\underline{u},\overline{u}]}U_n(u)-U(u)>\epsilon)$ and
$P(\sup_{u\in [\underline{u},\overline{u}]}U(u)-U_n(u)>\epsilon)$ for each
$n,\epsilon,\underline{u},\overline{u}$. Interestingly these quantities, seen
as a function of $\epsilon$, can be easily inverted numerically into functions
of the probability level $\delta$. Although not explicit, they can be computed
and tabulated. We plot such expressions and compare them to the classical bound
$\sqrt{\frac{\ln(1/\delta)}{2n}}$ provided by Massart inequality. Last, we
extend the local concentration results holding individually for each $n$ to
time-uniform concentration inequalities holding simultaneously for all $n$,
revisiting a reflection inequality by James, which is of independent interest
for the study of sequential decision making strategies.
