Tail Behaviour for Suprema of Empirical Processes

Abstract

We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ and appropriate constants $c, C$. When $k = 2$ these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general $k$ the bound can be used to obtain sharp upper-lower class results for the growth of $\sup_tX_n(t)$ with $n$.