ÓÒØÒØ× 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 v

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