Project

Spectral tail process; Heavy-tailed time series; Sojourn times ; Weighted Kolmogorov test; Ruin probability; Rare events and extremes ; Max-stable random fields; Multi-valued random fields; Queueing theory; Risk theory; Excursion sets; Ruin time approximation; Extremal index ; Cummulutive Parisian ruin; Uniform double-sum method; Fractonal Brownian motion on spheres; Berman type constants; Gaussian random fields

Lead
Rare Ereignisse und Extremewerte multidimensionaler Zufallsfelder
Lay summary
Wichtige Klimadaten wie  Lufttemperatur, Niederschlagsniveaus und Windgeschwindigkeiten gemessen an verschiedenen Stationen sowie Daten von Beobachtungen aus verschiedenen Galaxien sind typische Beispiele von multidimensionalen Daten in Zeit und Raum.  Seltene und extreme Ereignisse sind entscheidend für das Verständnis von Trends und zugrunde liegenden Phänomenen, welche anhand den obigen genannten Daten zur Analyse stehen. Die traditionelle Analyse von raren Ereignissen und Extremen beschäftigt sich mit der Modellierung von Frequenzen seltener Ereignisse und die Severitäten der Extreme. In den mathematischen Modellen, die in diesem Projekt untersucht werden sollen, wird die Gesamtzeit der seltenen Ereignisse, die durch Extremwerte (oder hohe Schwellenwerte) verursacht werden, als die "Sojourn time" bezeichnet. Die bisherige Literatur verfügt nur über Ad-hoc Methoden, welche teilweise geignet sind, "Sojourn time" oder "occupation time" zu analysiren. Das Hauptziel dieses Projektes ist die Entwicklung der theoretischen Methodik für die probabilistische Analyse der Sojourn-time und measure of excurssion sets.

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Last update: 03.10.2017

Natural persons

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Climate data such as air temperature, precipitation levels, and wind speeds measured at different locations on earth are typical examples of observations in time and space that can be modelled by multi-valued random fields. Other examples are total loss amounts from different insurance portfolios, the level of overloads of different queues, observation data from different galaxies, etc. Rare and extreme events are crucial for understanding trends and underlying phenomena, say global warming, or the occurrence of natural catastrophes. Traditional analysis of rare events and extremes is concerned with the modelling of frequencies of rare events and the severities of extremes. Advanced extreme value models and applications investigate further the total occurrence time of rare events which is essential for quantifying say climate changes, or for measuring the performance of stressed multiple queues. Taking again an example from climate data, observing in Switzerland air temperatures higher than 41 degrees or less than -41 degrees is quite rare, but have been already recorded! The local impact of such extremes strongly depends on the total duration in time of those occurrences. If such temperatures are observed say for only few minutes, their impact is negligible compared to extreme air temperatures observed over months or even years. In the mathematical models to be investigated in this paper, the total time of rare events caused by extreme values (or high thresholds) is referred to as the sojourn time. Observing -41.8 degrees in La Brévine on January 12, 1987 has only a local impact; occurrences of such extreme air temperatures in the whole Switzerland has much severe global impacts. Both the time and space that give rise to exceedances of high thresholds (referred to as the excursion set) pin down essential behaviours of rare events. Probabilistic analysis of sojourn times and volumes of excursion sets can be simplified by considering high thresholds.Currently in the literature, there are no developed methodologies or ad hoc results for theprobabilistic study of sojourn times and volumes of excursion sets of non-smooth multi-valued random processes and random fields, respectively. In order to analyse the probabilistic laws underlying the extreme air temperatures and extreme wind speeds, the number of observation should be considerably large. For such cases, extreme value theory justifies the use of max-stable processes and random fields as adequate limit models for spatial and temporal extremes.The main objective of this project is the development of the theoretical methodology for the probabilistic analysis of sojourn times/ volume of excursion sets of both stationary and non-stationary multi-valued random processes/fields with trend. Our asymptotic analysis will be primarily based on the {\it uniform double-sum method} for general homogeneous functionals that will be developed in this project. Additionally, this proposal includes the investigation of tractable max-stable processes/ random fields and discusses both theoretical and applied aspects related to their representations and max-domains of attraction. Further, our study includes characterisations of spectral tail processes of stationary heavy-tailed time series that are of interest for both max-stable processes and for the investigation of various asymptotic constants related to sojourn times of multi-valued random processes. Besides, this project will be concerned with the asymptotic analysis of specific complex models of rare event. In particular we shall be concerned with the extremal behaviour of Gaussian random fields on spheres, the tail behaviour of weighted Kolmogorov-Smirinov test statistics, approximation of ruin probabilities, exit and ruin times, and the performance of fluid multi-valued Gaussian queues.