A Note on Second Order Conditions in Extreme Value Theory: Linking General and Heavy Tail Conditions

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 ρe 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.

Keywords

• extreme value index
• regular variation
• semi-parametric estimation