On Compound Variables and the Asymptotic Behaviour of Their Tails

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Title: On Compound Variables and the Asymptotic Behaviour of Their Tails
Author: Tolvanen, Tuuli
Other contributor: Helsingin yliopisto, Matemaattis-luonnontieteellinen tiedekunta
University of Helsinki, Faculty of Science
Helsingfors universitet, Matematisk-naturvetenskapliga fakulteten
Publisher: Helsingin yliopisto
Date: 2020
Language: eng
URI: http://urn.fi/URN:NBN:fi:hulib-202008173791
Thesis level: master's thesis
Discipline: Soveltava matematiikka
Abstract: The objective of this thesis is to introduce the concept of compound variables and explain their use in one application specifically, as the total claim amount of an insurance company can be viewed as a compound variable. We study both the average behaviour as well as the tail behaviour of compound variables. Before delving into the results concerning the tails of compound variables, we aim to present an overview about the general theory and treat the average behaviour of compound variables first. We familiarize the reader with rudimentary concepts such as moment and cumulant generating functions. Along the way, the reader will also gain an understanding of both mixed variables as well as compound mixed variables. We state and prove some fundamental results concerning the expectation, variance and moment generating functions of compound variables. When the concept of compound variable is used to interpret the total claim amount, we also find the number of claims to be of interest. Since it is a random variable, we wish to be able to model it somehow. In the case of a general compound variable, the number of claims simply corresponds to the number of summands in the variable. We consider compound Poisson variables as a special case of compound variables. The reason for this is that if the counting variable or the number of claims variable is Poisson distributed, then the compound variable is a compound Poisson random variable. We also enhance the modelling of the number of claims by presenting mixing variables into the model. As a more general version for determining the expectation of a random sum we prove Wald's identity. It does not assume the independence of the counting variable and the increments in the same way we do in the definition of a compound variable. Towards the end, we shift the focus from general theory and average behaviour to tail behaviour of compound variables. We introduce the reader to the necessary classes of heavy-tailed and subexponential distributions to be able to formulate a few results that give an asymptotically equivalent approximation for the tail function of the compound variable. We prove the result for the case of the negative expectation of the increments (summands). We also present results for the case of non-negative expectation of the increments. Such a situation would be of interest in particular for total claim amounts, if we assume the claims being non-negative random variables.
Subject: Heavy-tails
Compound variables
Wald’s identity

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