GENERALIZED REGULAR VARIATION OF 2ND-ORDER

Assume that for a measurable function f on (0, infinity) there exist a positive auxiliary function a(t) and some gamma is an element of R su ch that phi(x) = lim(t-->infinity)(f(tx) - f(t))/a(t) = integral x-1 s (gamma-1)ds, x > 0. Then f is said to be of generalized regular variat ion. In order to control the asymptotic behaviour of certain estimator s for distributions in extreme value theory we are led to study regula r variation of second order, that is, we assume that lim(t-->infinity) (f(tx) - f(t) - a(t)phi(x))/a(1)(t) exists non-trivially with a second auxiliary function a(1)(t). We study the possible limit functions in this limit relation (defining generalized regular variation of second order) and their domains of attraction. Furthermore we give the corres ponding relation for the inverse function of a monotone f with the sta ted property. Finally, we present an Abel-Tauber theorem relating thes e functions and their Laplace transforms.