  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.
