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) ^2 (k-1) e^-2² P\ₜ XF (t) > \ C (k) ^2 (k-1) e^-2² for large 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.