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In this thesis we contribute to the extreme value theory by deriving conditional limit statements for bivariate random vectors with polar representations given that one component becomes extreme. Our results extend the class of elliptical and generalized distributions for which the conditional extreme value theory could be applied. For this purpose we propose a new geometric approach which permits deeper understanding of existing results and gives opportunities for our further interesting generalizations. Moreover, we deduce limit theorems for dependent polar components which are of much importance for applications., Diese Arbeit liefert einen Beitrag zur Extremwerttheorie, indem wir Grenzwertaussagen für die bedingte Verteilung von bivariaten Zufallsvektoren mit polarer Darstellung herleiten gegeben dass eine Vektorkomponente extrem wird. Unsere Ergebnisse erweitern die Klasse der elliptischen und verallgemeinerten Verteilungen, für die sich Aussagen über ihr asymptotisches Verhalten gewinnen lassen. Hierfür schlagen wir einen neuen geometrischen Ansatz vor, welcher sowohl ein tiefergehendes Verständnis von bestehenden Resultaten erlaubt als auch weitere interessante Verallgemeinerungen in unserer Arbeit ermöglicht. Darüber hinaus leiten wir Grenzwertaussagen für abhängige Polarkomponenten her, welche im Hinblick auf Anwendungen von großer Bedeutung sind.

In this thesis we contribute to the extreme value theory by deriving conditional limit statements for bivariate random vectors with polar representations given that one component becomes extreme. Our results extend the class of elliptical and generalized distributions for which the conditional extreme value theory could be applied. For this purpose we propose a new geometric approach which permits deeper understanding of existing results and gives opportunities for our further interesting generalizations. Moreover, we deduce limit theorems for dependent polar components which are of much importance for applications., Diese Arbeit liefert einen Beitrag zur Extremwerttheorie, indem wir Grenzwertaussagen für die bedingte Verteilung von bivariaten Zufallsvektoren mit polarer Darstellung herleiten gegeben dass eine Vektorkomponente extrem wird. Unsere Ergebnisse erweitern die Klasse der elliptischen und verallgemeinerten Verteilungen, für die sich Aussagen über ihr asymptotisches Verhalten gewinnen lassen. Hierfür schlagen wir einen neuen geometrischen Ansatz vor, welcher sowohl ein tiefergehendes Verständnis von bestehenden Resultaten erlaubt als auch weitere interessante Verallgemeinerungen in unserer Arbeit ermöglicht. Darüber hinaus leiten wir Grenzwertaussagen für abhängige Polarkomponenten her, welche im Hinblick auf Anwendungen von großer Bedeutung sind.

The global minimum variance portfolio (GMVP) is the starting point of the Markowitz mean-variance efficient frontier. The estimation of the GMVP weights is therefore of much importance for financial investors. The GMVP weights depend only on the inverse covariance matrix of returns on financial risky assets, for this reason the estimated GMVP weights are subject to substantial estimation risk, especially in high-dimensional portfolio settings. In this paper we review the recent literature on traditional sample estimators for the unconditional GMVP weights which are typically based on daily asset returns, as well as on modern realized estimators for the conditional GMVP weights based on intraday high-frequency returns. We present various types of GMVP estimators with the corresponding stochastic results, discuss statistical tests and point on some directions for further research. Our empirical application illustrates selected properties of realized GMVP weights. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical Models > Multivariate Models

The global minimum variance portfolio (GMVP) is the starting point of the Markowitz mean-variance efficient frontier. The estimation of the GMVP weights is therefore of much importance for financial investors. The GMVP weights depend only on the inverse covariance matrix of returns on financial risky assets, for this reason the estimated GMVP weights are subject to substantial estimation risk, especially in high-dimensional portfolio settings. In this paper we review the recent literature on traditional sample estimators for the unconditional GMVP weights which are typically based on daily asset returns, as well as on modern realized estimators for the conditional GMVP weights based on intraday high-frequency returns. We present various types of GMVP estimators with the corresponding stochastic results, discuss statistical tests and point on some directions for further research. Our empirical application illustrates selected properties of realized GMVP weights. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Multivariate Analysis Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data Statistical Models > Multivariate Models