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3044 results in Probability theory and stochastic processes

Page 74 of 153

Regular Variation
  • N. H. Bingham, C. M. Goldie, J. L. Teugels
  • Published online:
    05 July 2013
    Print publication:
    11 June 1987
  • This book is a comprehensive account of the theory and applications of regular variation. It is concerned with the asymptotic behaviour of a real function of a real variable x which is 'close' to a power of x. Such functions are much more than a convenient extension of powers. In many limit theorems regular variation is intrinsic to the result, and exactly characterises the limit behaviour. The book emphasises such characterisations, and gives a comprehensive treatment of those applications where regular variation plays an essential (rather then merely convenient) role. The authors rigorously develop the basic ideas of Karamata theory and de Haan theory including many new results and 'second-order' theorems. They go on to discuss the role of regular variation in Abelian, Tauberian, and Mercerian theorems. These results are then applied in analytic number theory, complex analysis, and probability, with the aim above all of setting the theory in context. A widely scattered literature is thus brought together in a unified approach. With several appendices and a comprehensive list of references, analysts, number theorists, and probabilists will find this an invaluable and complete account of regular variation. It will provide a rigorous and authoritative introduction to the subject for research students in these fields.

Stochastic Differential Equations on Manifolds
  • K. D. Elworthy
  • Published online:
    05 May 2013
    Print publication:
    02 September 1982
  • The aims of this book, originally published in 1982, are to give an understanding of the basic ideas concerning stochastic differential equations on manifolds and their solution flows, to examine the properties of Brownian motion on Riemannian manifolds when it is constructed using the stochiastic development and to indicate some of the uses of the theory. The author has included two appendices which summarise the manifold theory and differential geometry needed to follow the development; coordinate-free notation is used throughout. Moreover, the stochiastic integrals used are those which can be obtained from limits of the Riemann sums, thereby avoiding much of the technicalities of the general theory of processes and allowing the reader to get a quick grasp of the fundamental ideas of stochastic integration as they are needed for a variety of applications.

Probability, Statistics and Analysis
  • Edited by J. F. C. Kingman, G. E. H. Reuter
  • Published online:
    05 April 2013
    Print publication:
    10 February 1983
  • This collection of papers is dedicated to David Kendall (Professor of Mathematical Statistics in the University of Cambridge) on the occasion of his 65th birthday. The content of the contributions indicates the breadth of his interests in mathematics and statistics, and the interrelation between mathematical analysis, the theory of probability, and mathematical statistics. The topics will interest postgraduate and research mathematicians.

Brownian Motion, Hardy Spaces and Bounded Mean Oscillation
  • K. E. Petersen
  • Published online:
    05 April 2013
    Print publication:
    19 May 1977
  • This exposition of research on the martingale and analytic inequalities associated with Hardy spaces and functions of bounded mean oscillation (BMO) introduces the subject by concentrating on the connection between the probabilistic and analytic approaches. Short surveys of classical results on the maximal, square and Littlewood-Paley functions and the theory of Brownian motion introduce a detailed discussion of the Burkholder-Gundy-Silverstein characterization of HP in terms of maximal functions. The book examines the basis of the abstract martingale definitions of HP and BMO, makes generally available for the first time work of Gundy et al. on characterizations of BMO, and includes a probabilistic proof of the Fefferman-Stein Theorem on the duality of H11 and BMO.

The Theory of Probability
  • Explorations and Applications
  • Santosh S. Venkatesh
  • Published online:
    05 December 2012
    Print publication:
    08 November 2012
  • From classical foundations to advanced modern theory, this self-contained and comprehensive guide to probability weaves together mathematical proofs, historical context and richly detailed illustrative applications. A theorem discovery approach is used throughout, setting each proof within its historical setting and is accompanied by a consistent emphasis on elementary methods of proof. Each topic is presented in a modular framework, combining fundamental concepts with worked examples, problems and digressions which, although mathematically rigorous, require no specialised or advanced mathematical background. Augmenting this core material are over 80 richly embellished practical applications of probability theory, drawn from a broad spectrum of areas both classical and modern, each tailor-made to illustrate the magnificent scope of the formal results. Providing a solid grounding in practical probability, without sacrificing mathematical rigour or historical richness, this insightful book is a fascinating reference and essential resource, for all engineers, computer scientists and mathematicians.

  • Preface

  • Santosh S. Venkatesh, University of Pennsylvania
  • Book:
    The Theory of Probability
    Published online:
    05 December 2012
    Print publication:
    08 November 2012, pp xv-xxiv

  • Summary

    Gentle Reader: Henry Fielding begins his great comic novel Tom Jones with these words.

    An author ought to consider himself, not as a gentleman who gives a private or eleemosynary treat, but rather as one who keeps a public ordinary, at which all persons are welcome for their money. […] Men who pay for what they eat, will insist on gratifying their palates, however nice and even whimsical these may prove; and if every thing is not agreeable to their taste, will challenge a right to censure, to abuse, and to d—n their dinner without controul.

    To prevent therefore giving offence to their customers by any such disappointment, it hath been usual, with the honest and well-meaning host, to provide a bill of fare, which all persons may peruse at their first entrance into the house; and, having thence acquainted themselves with the entertainment which they may expect, may either stay and regale with what is provided for them, or may depart to some other ordinary better accommodated to their taste.

    To take a hint from these honest victuallers, as Fielding did, it strikes me therefore that I should at once and without delay explain my motivations for writing this book and what the reader may reasonably hope to find in it. To the expert reader who finds a discursive prolegomenon irritating, I apologise. There have been so many worthy and beautiful books published on the subject of probability that any new entry must needs perhaps make a case for what is being added to the canon.

Page 74 of 153