Milinčević, Nikolina (2023). Regular variation on Polish spaces, continuous maps and compound maxima. (Thesis). Universität Bern, Bern
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Abstract
This thesis focuses on regular variation in Polish spaces equipped with the general notion of scaling, bornology, and modulus. The bornology represents the collection of bounded sets, while the modulus generalises the metric to an arbitrary continuous Borel homogeneous function. We define and characterise regular variation and give numerous examples that show how the choice of scaling and bornology affects this definition. The main part of the thesis concerns continuous maps of regularly varying elements. We examine continuous bornologically consistent morphisms and show that, in many cases, they preserve the regular variation property. We start with a regularly varying random element ξ with a tail measure μ, defined on the Polish space X endowed with continuous scaling and topologically and scaling consistent bornology with countable base. We then map it continuously by a bornologically consistent morphism ψ to a space Y with the same properties as X. We show that if ψμ is nontrivial on the bornology on Y that satisfies given properties, then ψξ is regularly varying in Y. This result is further applied to the polar decomposition map and quotient mapping, among others. As in some simpler spaces, we decompose ξ into an ”angular” part that belongs to the set of all points with modulus one, and into a ”modular” part and show that this modular part is also regularly varying. Conditionally on modulus being large, we obtain a nontrivial limit of their joint distribution. This limit is expressed as a product measure of the Pareto-type measure and the spectral measure of ξ. For the continuous quotient map, we show that even though ξ is regularly varying, its equivalent class as a closed subset of X can contain both regularly varying and non-regularly varying selections. We show that if the selection map is chosen to be homogeneous with respect to scaling and if it minimises the modulus of the whole equivalence class, then the selected element is also regularly varying. In the last chapter, we consider an independent and identically distributed sequence (Xn) of random variables that belong to some maximum domain of attraction. We consider H = max{Xi : i = 1, 2, . . . ,K}, where K is a positive random integer with finite expectation. We show that as long as K is a stopping time with respect to the filtration generated by the sequence (Xn) and possibly some random element independent of the sequence (Xn), then H is in the same maximum domain of attraction as X1. We apply this result to some marked renewal cluster processes where we observe the maximum of all the observations that arrived until moment t > 0. Since this problem often arises in insurance models, it can be interpreted as a maximal claim problem until time t. We show that after proper normalisation, the maximal claim until time t converges in distribution to the distribution G if X1 is in the maximum domain of attraction of distribution G.
| Item Type: | Thesis |
|---|---|
| Dissertation Type: | Single |
| Date of Defense: | 4 July 2023 |
| Subjects: | 500 Science > 510 Mathematics |
| Institute / Center: | 08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science |
| Depositing User: | Sarah Stalder |
| Date Deposited: | 18 Jul 2023 12:52 |
| Last Modified: | 18 Jul 2023 12:52 |
| URI: | https://boristheses.unibe.ch/id/eprint/4433 |
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