| Abstract [eng] |
Over the years, heavy-tailed distributions have received considerable attention in a wide range of fields and are often used to model relatively rare but extreme events that classical distributions tend to underestimate. Since heavy-tailed behaviour is governed by large values in the tails, tail asymptotics becomes a vital characteristic. Due to the complex nature of systems in which heavy-tailed distributions are applied, various transformations of such distributions are often considered. In this dissertation we explore the asymptotic properties of three distinct transformations of heavy-tailed distributions. Firstly, we investigate the tail probability asymptotics of the product of independent zero-mean normal random variables, which exhibit heavy-tailed behaviour when at least three multipliers are involved. The obtained results provide a more precise asymptotic formula with the first leading-order error term, thus offering more insight into the decay rate of the tail probability. Secondly, we explore the tail asymptotics of the generalised tail moments of randomly weighted sums with dominatedly varying and not necessarily independent random summands under a measurable transformation satisfying certain properties eventually. The main findings provide asymptotic bounds given by the sums of generalised tail moments of transformed individual summands with the specific correcting constants. Thirdly, we establish asymptotic formulas of the Haezendonck-Goovaerts risk measure specified by a power Young function for the randomly weighted sums with consistently varying and possibly dependent random summands when the confidence level of the measure tends to unit. The main results enable to asymptotically express the risk measure for a sum of random risks as a certain functional dependent on the marginal distributions of individual risks. It follows, that to find the asymptotic estimate there is no need to evaluate complex distributions of the sum of random variables. The main findings of the dissertation are derived analytically, using the classical results from probability theory and real analysis as well as the methods from extreme value theory. To illustrate and empirically validate the obtained theoretical results, Monte Carlo simulation studies are provided. |
