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.