Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings. (March 2015) - Kosiński, K. M.; Mandjes, M.
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.