I encountered a problem while reading a classic paper. The random variable $V \sim exp(1)$, $(v_{(1)},v_{(2)},...,v_{(n)})$ are ordered. The paper said, for any $\epsilon$, based on the Kolmogorov inequality, there exist $v_{in}$ and $v^{in}$,$$sup_{1\leq i\leq n}[v^{in}-v_{in}]\leq K(\epsilon)$$ and $$ P\left\{v_{i n}(\epsilon)<v_{(i )}<v^{i n}(\epsilon), 1 \leq i \leq n\right\} \geq 1-\epsilon $$
It is the proposition 4 of this paper(p56). It doesn’t seem to be difficult from the writing tone, but I cannot understand. And I think it is not the maximal inequality which is called as Kolmogorov inequality.
Thank you!