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Asymptotic Expansions for Infinite Weighted Convolutions of Heavy Tail Distributions and Applications
Ph. Barbe, CNRS, Paris, France, and W. P. McCormick, University of Georgia, Athens
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Memoirs of the American Mathematical Society
2009; 117 pp; softcover
Volume: 197
ISBN-10: 0-8218-4259-5
ISBN-13: 978-0-8218-4259-1
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/197/922
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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

  • Introduction
  • Main result
  • Implementing the expansion
  • Applications
  • Preparing the proof
  • Proof in the positive case
  • Removing the sign restriction on the random variables
  • Removing the sign restriction on the constants
  • Removing the smoothness restriction
  • Appendix. Maple code
  • Bibliography
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