# Asymptotics of Randomly Weighted Sums of Heavy-Tailed Random Variables

 dc.contributor Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta fi dc.contributor University of Helsinki, Faculty of Science en dc.contributor Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten sv dc.contributor.author Pyrylä, Atte dc.date.issued 2020 dc.identifier.uri URN:NBN:fi:hulib-202011254601 dc.identifier.uri http://hdl.handle.net/10138/321917 dc.description.abstract In this thesis we will look at the asymptotic approach to modeling randomly weighted heavy-tailed random variables and their sums. The heavy-tailed distributions, named after the defining property of having more probability mass in the tail than any exponential distribution and thereby being heavy, are essentially a way to have a large tail risk present in a model in a realistic manner. The weighted sums of random variables are a versatile basic structure that can be adapted to model anything from claims over time to the returns of a portfolio, while giving the primary random variables heavy-tails is a great way to integrate extremal events into the models. The methodology introduced in this thesis offers an alternative to some of the prevailing and traditional approaches in risk modeling. en Our main result that we will cover in detail, originates from "Randomly weighted sums of subexponential random variables" by Tang and Yuan (2014), it draws an asymptotic connection between the tails of randomly weighted heavy-tailed random variables and the tails of their sums, explicitly stating how the various tail probabilities relate to each other, in effect extending the idea that for the sums of heavy-tailed random variables large total claims originate from a single source instead of being accumulated from a bunch of smaller claims. A great merit of these results is how the random weights are allowed for the most part lack an upper bound, as well as, be arbitrarily dependent on each other. As for the applications we will first look at an explicit estimation method for computing extreme quantiles of a loss distributions yielding values for a common risk measure known as Value-at-Risk. The methodology used is something that can easily be adapted to a setting with similar preexisting knowledge, thereby demonstrating a straightforward way of applying the results. We then move on to examine the ruin problem of an insurance company, developing a setting and some conditions that can be imposed on the structures to permit an application of our main results to yield an asymptotic estimate for the ruin probability. Additionally, to be more realistic, we introduce the approach of crude asymptotics that requires little less to be known of the primary random variables, we formulate a result similar in fashion to our main result, and proceed to prove it. dc.language.iso eng dc.publisher Helsingin yliopisto fi dc.publisher University of Helsinki en dc.publisher Helsingfors universitet sv dc.subject Asymptotics dc.subject Randomly weighted sum dc.subject Risk modeling dc.subject Subexponential distribution dc.title Asymptotics of Randomly Weighted Sums of Heavy-Tailed Random Variables en dc.type.ontasot pro gradu -tutkielmat fi dc.type.ontasot master's thesis en dc.type.ontasot pro gradu-avhandlingar sv dc.subject.discipline none und dct.identifier.urn URN:NBN:fi:hulib-202011254601 dc.subject.specialization Sovellettu matematiikka fi dc.subject.specialization Applied Mathematics en dc.subject.specialization Tillämpad matematik sv dc.subject.degreeprogram Matematiikan ja tilastotieteen maisteriohjelma fi dc.subject.degreeprogram Master's Programme in Mathematics and Statistics en dc.subject.degreeprogram Magisterprogrammet i matematik och statistik sv