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

TL;DR

This paper develops a theoretical framework for the extreme value analysis of all-time suprema of aggregated self-similar Gaussian processes with trend, revealing convergence to various limiting distributions depending on model parameters.

Contribution

It introduces new limit theorems for normalized order statistics of Gaussian process suprema with trend, extending extreme value theory to non-stationary sequences.

Findings

Normalized order statistics converge to Erlang, Normal, or mixed distributions.

Moments of normalized statistics converge to those of the limiting distribution.

Results apply to both stationary and non-stationary Gaussian sequences.

Abstract

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.