Extremes of normed empirical moment generating function processes

Michael Stewart, John Robinson

Abstract

We present a method for deriving the limiting distribution of the maximum of a normed empirical moment generating function process indexed by one parameter. We first extend slightly the results of Csörgő, Csörgő, Horváth and Mason (1986) to provide the rate of convergence for a Gaussian approximation to a non-Donsker empirical process. In cases we consider, the maximum tends to infinity in probability, but when appropriately scaled has a limiting Gumbel extreme value distribution.

Keywords: Empirical processes, extreme value distribution, limiting distribution, empirical moment generating function, extremes of Gaussian processes.

AMS Subject Classification: Primary 62E20; secondary 60G70.