LIMIT THEOREMS FOR RANDOM EXPONENTIAL SUMS AND THEIR APPLICATIONS TO INSURANCE AND THE RANDOM ENERGY MODEL

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Erturk, H. (2016). LIMIT THEOREMS FOR RANDOM EXPONENTIAL SUMS AND THEIR APPLICATIONS TO INSURANCE AND THE RANDOM ENERGY MODEL. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

In this dissertation, we are mainly concerned with the sum of random exponentials. Here, the random variables are independent and identically distributed. Another distinctive assumption is the number of variables in this sum is a function of the constant on the exponent. Our first goal is to find the limiting distributions of the random exponential sums for new class of the random variables. For some classes, such results are known; normal distribution, Weibull distribution etc. Secondly, we apply these limit theorems to some insurance models and the random energy model in statistical physics. Specifically for the first case, we give the estimate of the ruin probability in terms of the empirical data. For the random energy model, we present the analysis of the free energy for new class of distribution. In some particular cases, we prove the existence of several critical points for the free energy. In some other cases, we prove the absence of phase transitions. Our results give a new approach to compute the ruin probabilities of insurance portfolios empirically when there is a sequence of insurance portfolios with a custom growth rate of the claim amounts. The second application introduces a simple method to drive the free energy in the case the random variables in the statistical sum can be represented as a function of standard exponential random variables. The technical tool of this study includes the classical limit theory for the sum of independent and identically distributed random variables and different asymptotic methods like the Euler-Maclaurin formula and Laplace method.

Details
Author

Erturk, Huseyin

Title

LIMIT THEOREMS FOR RANDOM EXPONENTIAL SUMS AND THEIR APPLICATIONS TO INSURANCE AND THE RANDOM ENERGY MODEL

Physical Description

1 online resource (70 pages) : PDF

Date

2016

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

In this dissertation, we are mainly concerned with the sum of random exponentials. Here, the random variables are independent and identically distributed. Another distinctive assumption is the number of variables in this sum is a function of the constant on the exponent. Our first goal is to find the limiting distributions of the random exponential sums for new class of the random variables. For some classes, such results are known; normal distribution, Weibull distribution etc. Secondly, we apply these limit theorems to some insurance models and the random energy model in statistical physics. Specifically for the first case, we give the estimate of the ruin probability in terms of the empirical data. For the random energy model, we present the analysis of the free energy for new class of distribution. In some particular cases, we prove the existence of several critical points for the free energy. In some other cases, we prove the absence of phase transitions. Our results give a new approach to compute the ruin probabilities of insurance portfolios empirically when there is a sequence of insurance portfolios with a custom growth rate of the claim amounts. The second application introduces a simple method to drive the free energy in the case the random variables in the statistical sum can be represented as a function of standard exponential random variables. The technical tool of this study includes the classical limit theory for the sum of independent and identically distributed random variables and different asymptotic methods like the Euler-Maclaurin formula and Laplace method.

Genre

doctoral dissertations

Subjects--Topics

Mathematics
Statistics

Degree

Ph.D.

Subject Area

Mathematics

Advisor(s)

Molchanov, Stanislav

Committee Members

Sonin, Isaac
Grabchak, Michael
Allan, Craig

Degree Note

Thesis (Ph.D.)--University of North Carolina at Charlotte, 2016.

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Rights Holder Information

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Identifier

Erturk_uncc_0694D_11038

Permalink

http://hdl.handle.net/20.500.13093/etd:763