  The seminal papers of Pickands [1,2] paved the way for a systematic study of
high exceedance probabilities of both stationary and non-stationary Gaussian
processes. Yet, in the vector-valued setting, due to the lack of key tools
including Slepian's Lemma, Borell-TIS and Piterbarg inequalities there has not
been any methodological development in the literature for the study of extremes
of vector-valued Gaussian processes. In this contribution we develop the
uniform double-sum method for the vector-valued setting obtaining the exact
asymptotics of the exceedance probabilities for both stationary and
non-stationary Gaussian processes. We apply our findings to the operator
fractional Brownian motion and the operator fractional Ornstein-Uhlenbeck
process.
