On Weak Convergence of Extremal Processes

Abstract

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.