A note on weak convergence to extremal processes

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