Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

Abstract

Let W = { W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we find that logP(there exists n ∈ N: W n u q ) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {a n }, {v n }, (1 / v n )logP( W n / a n ≥ u q ) has a, continuous in q , limit J W ( q ). We allow the scalings {a n } and {v n } to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n / a n satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.