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

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*Ph. Barbe; W. P. McCormick*

The authors establish some asymptotic expansions for infinite weighted
convolution of distributions having regularly varying tails.
Applications to linear time series models, tail index estimation,
compound sums, queueing theory, branching processes, infinitely
divisible distributions and implicit transient renewal equations are
given.

A noteworthy feature of the approach taken in this paper is that
through the introduction of objects, which the authors call the
Laplace characters, a link is established between tail area expansions
and algebra. By virtue of this representation approach, a unified
method to establish expansions across a variety of problems is
presented and, moreover, the method can be easily programmed so that a
computer algebra package makes implementation of the method not only
feasible but simple.

#### Table of Contents

# Table of Contents

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

- Contents v6 free
- 1. Introduction 19 free
- 2. Main result 917
- 3. Implementing the expansion 2129
- 4. Applications 3947
- 5. Preparing the proof 6573
- 6. Proof in the positive case 7583
- 7. Removing the sign restriction on the random variables 97105
- 8. Removing the sign restriction on the constants 105113
- 9. Removing the smoothness restriction 109117
- Appendix. Maple code 111119
- Bibliography 115123