Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications.

Intro -- Contents -- 1. Introduction -- 1.1. Prolegomenom -- 1.2. Mathematical overview and heuristics -- 2. Main result -- 2.1. Some notation -- 2.2. Asymptotic scales -- 2.3. The Laplace characters -- 2.4. Smoothly varying functions of finite order -- 2.5. Asymptotic expansion for in finite weighted convolution -- 3. Implementing the expansion -- 3.1. How many terms are in the expansion? -- 3.2. [sub(*)]-Asymptotic scales and functions of class m -- 3.3. Tail calculus: From Laplace characters to linear algebra -- 3.4. Some examples -- 3.5. Two terms expansion and second order regular variation -- 3.6. Some open questions -- 4. Applications -- 4.1. ARMA models -- 4.2. Tail index estimation -- 4.3. Randomly weighted sums -- 4.4. Compound sums -- 4.5. Queueing theory -- 4.6. Branching processes -- 4.7. Infinitely divisible distributions -- 4.8. Implicit transient renewal equation and iterative systems -- 5. Preparing the proof -- 5.1. Properties of Laplace characters -- 5.2. Properties of smoothly varying functions of finite order -- 6. Proof in the positive case -- 6.1. Decomposition of the convolution into integral and multiplication operators -- 6.2. Organizing the proof -- 6.3. Regular variation and basic tail estimates -- 6.4. The fundamental estimate -- 6.5. Basic lemmas -- 6.6. Inductions -- 6.7. Conclusion -- 7. Removing the sign restriction on the random variables -- 7.1. Elementary properties of U[sub(H)] -- 7.2. Basic expansion of U[sub(H)] -- 7.3. A technical lemma -- 7.4. Conditional expansion and removing conditioning -- 8. Removing the sign restriction on the constants -- 8.1. Neglecting terms involving the multiplication operators -- 8.2. Substituting H[sup((k))] and G[sup((k))] by their expansions -- 9. Removing the smoothness restriction -- Appendix. Maple code -- Bibliography.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.