Large deviations of bivariate Gaussian extrema

Harsha Honnappa,Remco Van Der Hofstad

doi:10.1007/s11134-019-09632-z

Abstract

We establish sharp tail asymptotics for componentwise 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 versus 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.

Highlights

Motivated by applications to the analysis of queueing networks, we study the large deviations of extreme values of multivariate Gaussians
We characterize the likelihood of the tail event {Xn > anu}, where u ∈ (0, ∞), as the number of random vectors n tends to infinity, under the assumption that an → ∞ as n → ∞
We prove a leading order asymptote for the extreme value that aligns with the result in case 1

Summary

Introduction

Motivated by applications to the analysis of queueing networks, we study the large deviations of extreme values of multivariate Gaussians. In all the cases where a term + 2 is present, the maximum is attained by two different indices of Xi , one which attains the maximum of the first coordinate and one which attains the maximum of the second coordinate. The latter case has the most distinct possibilities (“larger entropy”), while the first may have a larger probability for appropriate correlation coefficients ρ. In Theorem 4, on the other hand, we establish a sharp asymptote for the likelihood and show that there exists a continuous function I : Rd → R and constants K , b and c such that. The proofs of these theorems use the inclusion–exclusion principle to bound the likelihood from above and below

Related literature

Notation and setting

Right scale asymptote

Large-scale asymptote