Approximation of Kolmogorov-Smirnov Test Statistics

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 $W(\mathit{x})$ 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.