 | Barbe Ph. Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications / Ph.Barbe, W.P.McCormick. - Providence: American Mathematical Society, 2009. - vii, 117 p. - (Memoirs of the American Mathematical Society; Vol.197, N 922). - Bibliogr.: p.115-117. - ISBN 978-0-8218-4259-1; ISSN 0065-9266
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1. Introduction ................................................. 1
1.1. Prolegomenom ............................................ 1
1.2. Mathematical overview and heuristics .................... 4
2. Main result .................................................. 9
2.1. Some notation ........................................... 9
2.2. Asymptotic scales ...................................... 10
2.3. The Laplace characters ................................. 12
2.4. Smoothly varying functions of finite order ............. 15
2.5. Asymptotic expansion for infinite weighted
convolution ............................................ 16
3. Implementing the expansion .................................. 21
3.1. How many terms are in the expansion? ................... 21
3.2. *-Asymptotic scales and functions of class m ........... 24
3.3. Tail calculus: From Laplace characters to linear
algebra ................................................ 27
3.4. Some examples .......................................... 28
3.5. Two terms expansion and second order regular
variation .............................................. 34
3.6. Some open questions .................................... 36
4. Applications ................................................ 39
4.1. ARMA models ............................................ 39
4.2. Tail index estimation .................................. 40
4.3. Randomly weighted sums ................................. 47
4.4. Compound sums .......................................... 50
4.5. Queueing theory ........................................ 53
4.6. Branching processes .................................... 55
4.7. Infinitely divisible distributions ..................... 56
4.8. Implicit transient renewal equation and iterative
systems ................................................ 58
5. Preparing the proof ......................................... 65
5.1. Properties of Laplace characters ....................... 65
5.2. Properties of smoothly varying functions of finite
order .................................................. 67
6. Proof in the positive case .................................. 75
6.1. Decomposition of the convolution into integral and
multiplication operators ............................... 75
6.2. Organizing the proof ................................... 77
6.3. Regular variation and basic tail estimates ............. 79
6.4. The fundamental estimate ............................... 82
6.5. Basic lemmas ........................................... 83
6.6. Inductions ............................................. 89
6.7. Conclusion ............................................. 94
7. Removing the sign restriction on the random variables ....... 97
7.1. Elementary properties of UH ............................ 98
7.2. Basic expansion of UH .................................. 99
7.3. A technical lemma ..................................... 100
7.4. Conditional expansion and removing conditioning ....... 102
8. Removing the sign restriction on the constants ............. 105
8.1. Neglecting terms involving the multiplication
operators ............................................. 105
8.2. Substituting H(k) and G(k) by their expansions ......... 107
9. Removing the smoothness restriction ........................ 109
Appendix. Maple code .......................................... 111
Bibliography .................................................. 115
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