Tail Behaviour for Suprema of Empirical Processes

We consider multivariate empirical processes Xₙ(t) := √ n (Fₙ(t) - F(t)), where Fₙ is an empirical distribution function based on i.i.d. variables with distribution function F and t ∈ Rᵏ. For XF the weak limit of Xₙ, it is shown that c(F, k)λ²⁽ᵏ⁻¹⁾e-2λ² ≤ P\ₜ XF(t) > λ\ ≤ C(k)λ²⁽ᵏ⁻¹⁾e-2λ² for large λ 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 ₜXₙ(t) with n.