REVSTAT – Statistical Journal Volume 5, Number 3, November 2007, 285–304 A NOTE ON SECOND ORDER CONDITIONS IN EXTREME VALUE THEORY: LINKING GENERAL AND HEAVY TAIL CONDITIONS Authors: M. Isabel Fraga Alves – CEAUL, DEIO, Faculty of Science, University of Lisbon, Portugal isabel.alves@fc.ul.pt M. Ivette Gomes – CEAUL, DEIO, Faculty of Science, University of Lisbon, Portugal ivette.gomes@fc.ul.pt Laurens de Haan – Department of Economics, Erasmus University Rotterdam, The Netherlands ldhaan@few.eur.nl Cl´ audia Neves – UIMA, Department of Mathematics, University of Aveiro, Portugal claudia.neves@ua.pt Received: July 2007 Revised: October 2007 Accepted: October 2007 Abstract: • Second order conditions ruling the rate of convergence in any first order condition involving regular variation and assuring a unified extreme value limiting distribution function for the sequence of maximum values, linearly normalized, have appeared in several contexts whenever researchers are working either with a general tail, i.e., γ ∈ R, or with heavy tails, with an extreme value index γ> 0. In this paper we shall clarify the link between the second order parameters, say ρ and  ρ that have appeared in the two above mentioned set-ups, i.e., for a general tail and for heavy tails, respectively. We illustrate the theory with some examples and, for heavy tails, we provide a link with a third order framework. Key-Words: • extreme value index; regular variation; semi-parametric estimation. AMS Subject Classification: • Primary 62G32, 62E20, 26A12.