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

Ph. Barbe CNRS, Paris, France
W. P. McCormick University of Georgia, Athens
Available Formats:
Electronic ISBN: 978-1-4704-0528-1
Product Code: MEMO/197/922.E
List Price: $71.00 MAA Member Price:$63.90
AMS Member Price: $42.60 Click above image for expanded view Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications Ph. Barbe CNRS, Paris, France W. P. McCormick University of Georgia, Athens Available Formats:  Electronic ISBN: 978-1-4704-0528-1 Product Code: MEMO/197/922.E  List Price:$71.00 MAA Member Price: $63.90 AMS Member Price:$42.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1972009; 117 pp
MSC: Primary 41; 60; Secondary 44; 62;

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.

• Chapters
• 1. Introduction
• 2. Main result
• 3. Implementing the expansion
• 4. Applications
• 5. Preparing the proof
• 6. Proof in the positive case
• 7. Removing the sign restriction on the random variables
• 8. Removing the sign restriction on the constants
• 9. Removing the smoothness restriction
• Appendix. Maple code
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Accessibility – to request an alternate format of an AMS title
Volume: 1972009; 117 pp
MSC: Primary 41; 60; Secondary 44; 62;

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.

• Chapters
• 1. Introduction
• 2. Main result
• 3. Implementing the expansion
• 4. Applications
• 5. Preparing the proof
• 6. Proof in the positive case
• 7. Removing the sign restriction on the random variables
• 8. Removing the sign restriction on the constants
• 9. Removing the smoothness restriction
• Appendix. Maple code
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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