The strong approximation of extremal processes

Summary If X₁, X₂, ..., are i.i.d. random variables and Y_n=Max(X₁, ..., X_n); if for some sequences A_n, B_n, n=1, 2, ..., E_n(t)=A_nY_[nt]+B_n is such that E_n(1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E⁽ⁿ⁾(t) on the same probability space, such that d(E_n(.), E⁽ⁿ⁾(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations.