  We investigate extreme value theory of a class of random sequences defined by
the all-time suprema of aggregated self-similar Gaussian processes with trend.
This study is motivated by its potential applications in various areas and its
theoretical interestingness. We consider both stationary sequences and
non-stationary sequences obtained by considering whether the trend functions
are identical or not. We show that a sequence of suitably normalised $k$th
order statistics converges in distribution to a limiting random variable which
can be a negative log transformed Erlang distributed random variable, a Normal
random variable or a mixture of them, according to three conditions deduced
through the model parameters. Remarkably, this phenomenon resembles that for
the stationary Normal sequence. We also show that various moments of the
normalised $k$th order statistics converge to the moments of the corresponding
limiting random variable. The obtained results enable us to analyze various
properties of these random sequences, which reveals the interesting
particularities of this class of random sequences in extreme value theory.
