  Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in
this contribution, we concern the asymptotics of \begin{align*}
\mathbb{P}\left\{\sup_{\boldsymbol{x}\in [0,1]^n}\left(W(\boldsymbol{x})\Big|
W(\boldsymbol{1})=w\right)>u\right\}, \ w\in\mathbb{R}, \end{align*} for large
$u$ where $W(\boldsymbol{x})$ is the multivariate Brownian sheet based on a
distribution function $F$. The results related to general $F$ are investigated
and some important examples are also showed.

Probability