Large Deviations of Bivariate Gaussian Extrema

We establish sharp tail asymptotics for component-wise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle, and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different vs. the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues

Logarithmic asymptotics for multidimensional extremes under nonlinear scalings

Let W = {Wn: n ¿ N} be a sequence of random vectors in Rd, d = 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ¿ N: Wn u q) as u ¿ 8. 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 {an}, {vn}, (1 / vn)logP(Wn / an = u q) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature