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

Lanpeng Ji and Xiaofan Peng

Stochastic Processes and their Applications, 2023, vol. 158, issue C, 418-452

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 kth 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 kth order statistics converge to the moments of the corresponding limiting random variable. The obtained results enable us to analyse various properties of these random sequences, which reveals the interesting particularities of this class of random sequences in extreme value theory.

Keywords: Extreme value; Gaussian processes; Generalised Weibull-like distribution; Poisson convergence; Order statistics; Phantom distribution function (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1016/j.spa.2023.01.013

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