Extreme value theory for a sequence of supremum of a class of Gaussian processes with trend

We investigate extreme value theory of a stationary random sequence defined by the all-time supremum of aggregated self-similar Gaussian processes with trend. This study is mainly motivated by a recent contribution by Meijer et al. [Extreme-value theory for large fork-join queues, with applications to high-tech supply chains, 2021, preprint] where some extreme value theory for the Brownian case is discussed. We aim to extend some of their results on this theory. 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 of the stationary Normal sequence. We also show that various moments of the normalised $k$th order statistics converge to the corresponding moments of the limiting random variable.