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Asymptotic Analysis of Random Walks
Heavy-Tailed Distributions


Part of Encyclopedia of Mathematics and its Applications

  • Date Published: June 2008
  • availability: Available
  • format: Hardback
  • isbn: 9780521881173

£ 156.00

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About the Authors
  • This book focuses on the asymptotic behaviour of the probabilities of large deviations of the trajectories of random walks with 'heavy-tailed' (in particular, regularly varying, sub- and semiexponential) jump distributions. Large deviation probabilities are of great interest in numerous applied areas, typical examples being ruin probabilities in risk theory, error probabilities in mathematical statistics, and buffer-overflow probabilities in queueing theory. The classical large deviation theory, developed for distributions decaying exponentially fast (or even faster) at infinity, mostly uses analytical methods. If the fast decay condition fails, which is the case in many important applied problems, then direct probabilistic methods usually prove to be efficient. This monograph presents a unified and systematic exposition of the large deviation theory for heavy-tailed random walks. Most of the results presented in the book are appearing in a monograph for the first time. Many of them were obtained by the authors.

    • The first unified systematic exposition of the large deviations theory for heavy-tailed random walks
    • Computing the probabilities of these large deviations (rare events) is essential, in fields such as finance, networks and insurance; this books shows you how
    • Includes many never before published results on large deviations for random walks with non-identically distributed jumps
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    Reviews & endorsements

    'This book is a worthy tribute to the amazing fecundity of the structure of random walks!' Mathematical Reviews

    '… an up-to-date, unified and systematic exposition of the field. Most of the results presented are appearing in a monograph for the first time and a good proportion of them were obtained by the authors. … The book presents some beautiful and useful mathematics that may attract a number of probabilists to the large deviations topic in probability.' EMS Newsletter

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    Product details

    • Date Published: June 2008
    • format: Hardback
    • isbn: 9780521881173
    • length: 656 pages
    • dimensions: 242 x 162 x 48 mm
    • weight: 1.22kg
    • contains: 5 b/w illus.
    • availability: Available
  • Table of Contents

    1. Preliminaries
    2. Random walks with jumps having no finite first moment
    3. Random walks with finite mean and infinite variance
    4. Random walks with jumps having finite variance
    5. Random walks with semiexponential jump distributions
    6. Random walks with exponentially decaying distributions
    7. Asymptotic properties of functions of distributions
    8. On the asymptotics of the first hitting times
    9. Large deviation theorems for sums of random vectors
    10. Large deviations in the space of trajectories
    11. Large deviations of sums of random variables of two types
    12. Non-identically distributed jumps with infinite second moments
    13. Non-identically distributed jumps with finite variances
    14. Random walks with dependent jumps
    15. Extension to processes with independent increments
    16. Extensions to generalised renewal processes
    Bibliographic notes
    Index of notations

  • Authors

    A. A. Borovkov, Sobolev Institute of Mathematics, Novosibirsk
    Alexander Borovkov works at the Sobolev Institute of Mathematics in Novosibirsk.

    K. A. Borovkov, University of Melbourne
    Konstantin Borovkov is a staff member in the Department of Mathematics and Statistics at the University of Melbourne.

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