| Abstract [eng] |
Although this dissertation is primarily theoretical in nature, the relevance of heavy-tailed distributions extends well beyond purely theoretical considerations. Such distributions play an important role in financial and actuarial risk management, where extreme events, such as catastrophic losses or exceptionally large claims, cannot be regarded as negligible fluctuations. The dissertation is primarily concerned with the asymptotic analysis of random sums composed of heavy-tailed random variables. One of the main objectives is to generalize existing results on the asymptotic behaviour of generalized left truncated moments of random and randomly weighted sums with a fixed number of heavy-tailed summands by allowing a broader class of transformations. The research is further extended to randomly stopped sums, where inverse-type problems are investigated. Conditions are derived under which dominated variation or extended regularity of randomly stopped sums determines the stopping moment to belong to the class of dominatedly varying distributions. Finally, the dissertation examines the Haezendonck–Goovaerts risk measure for aggregate losses. The obtained results show that, under suitable conditions, the asymptotic behaviour of this risk measure is determined by the dominant summands, allowing substantial simplifications in risk evaluation. |
