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−2λ^2≤ P{suptXF(t) > λ} ≤ C(k) λ2(k−1)e−2λ^2 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 suptXn(t) with n.