L. Dehaan et U. Stadtmuller, GENERALIZED REGULAR VARIATION OF 2ND-ORDER, Journal of the Australian Mathematical Society. Series A. Pure mathematics and statistics, 61, 1996, pp. 381-395
Citations number
17
Categorie Soggetti
Mathematics, General","Statistic & Probability",Mathematics,"Statistic & Probability
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.