A uniform bound for the deviation of empirical distribution functions

If X1, ..., Xn are independent Rd-valued random vectors with common distribution function F, and if Fn is the empirical distribution function for X1, ..., Xn, then, among other things, it is shown that P{supx | Fn(x) | [greater-or-equal, slanted] [epsilon]} [less-than-or-equals, slant] 2e2(2n)de-2n[epsilon]2 for all n[epsilon]2 >= d2. The inequality remains valid if the Xi are not identically distributed and F(x) is replaced by [Sigma]iP{Xi <= x}/n.