Let {Xn, n [greater-or-equal, slanted] 1} be a sequence of independent random variables and Mn = max1 [less-than-or-equals, slant] k [less-than-or-equals, slant] n Xk, n [greater-or-equal, slanted] 1. Define for n [greater-or-equal, slanted] 1, Yn(t) = Gn-1(X1) if 0 < t < 1/n, = Gn-1(M[n]) if 1/n [less-than-or-equals, slant] t. where {Gn, n [greater-or-equal, slanted] 1} is a sequence of strictly increasing and continuous functions; and Gn-1 is the inverse of Gn. We show that if Yn(1) converges in distribution to a nondegenerate random variable then the process {Yn(t)} converges weakly to an extremal process under the Skorokhod J1-topology. The weak convergence is established when (i) the Xn's are identically distributed, and (ii) the distribution function (d.f.) of Xn is one of r distinct d.f.'s F1,...,Fr with some additional assumptions.
