# Large Deviations of Bivariate Gaussian Extrema

## TL;DR

This paper derives precise tail asymptotics for the joint extreme values of bivariate Gaussian vectors, revealing how correlation influences the likelihood of simultaneous maxima and providing insights relevant for network queue analysis.

## Contribution

It establishes sharp tail asymptotics and a large deviations principle for bivariate Gaussian extremes with arbitrary correlation, a novel extension in the field.

## Key findings

- Identifies conditions when maxima occur at different vs. same indices based on correlation.
- Provides explicit rate functions for tail asymptotics.
- Enhances understanding of Gaussian process extremes and their applications.

## Abstract

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.