Selected Works in Probability and Statistics

Selected Works of Donald L. Burkholder

Editors: Davis, Burgess, Song, Renming (Eds.)

  • Brings together the central research and expository papers of Donald Burkholder
  • Burkholder's research is important for its breadth, importance, and unity
  • Commentaries explain the development of Burkholder's ideas over time
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About this book

This book chronicles Donald Burkholder's thirty-five year study of martingales and its consequences. Here are some of the highlights. Pioneering work by Burkholder and Donald Austin on the discrete time martingale square function led to Burkholder and Richard Gundy's proof of inequalities comparing the quadratic variations and maximal functions of continuous martingales, inequalities which are now indispensable tools for stochastic analysis. Part of their proof showed how novel distributional inequalities between the maximal function and quadratic variation lead to inequalities for certain integrals of functions of these operators. The argument used in their proof applies widely and is now called the Burkholder-Gundy good lambda method. This uncomplicated and yet extremely elegant technique, which does not involve randomness, has become important in many parts of mathematics. The continuous martingale inequalities were then used by Burkholder, Gundy, and Silverstein to prove the converse of an old and celebrated theorem of Hardy and Littlewood. This paper transformed the theory of Hardy spaces of analytic functions in the unit disc and extended and completed classical results of Marcinkiewicz concerning norms of conjugate functions and Hilbert transforms. While some connections between probability and analytic and harmonic functions had previously been known, this single paper persuaded many analysts to learn probability. These papers together with Burkholder's study of martingale transforms led to major advances in Banach spaces. A simple geometric condition given by Burkholder was shown by Burkholder, Terry McConnell, and Jean Bourgain to characterize those Banach spaces for which the analog of the Hilbert transform retains important properties of the classical Hilbert transform. Techniques involved in Burkholder's usually successful pursuit of best constants in martingale inequalities have become central to extensive recent research into two well- known open problems, one involving the two dimensional Hilbert transform and its connection to quasiconformal mappings and the other a conjecture in the calculus of variations concerning rank-one convex and quasiconvex functions. This book includes reprints of many of Burkholder's papers, together with two commentaries on his work and its continuing impact.

About the authors

Burgess Davis is Professor of Statistics & Mathematics at Purdue University.

Renming Song is Professor in the Department of Mathematics at the University of Illinois at Urbana-Champaign.

Reviews

From the reviews:

“This book is one of Springer’s Selected Works in Probability and Statistics series offering to readers an opportunity of getting assembled and commented works of distinguished scholars in probability and statistics. … The reviewed book contains reprints of 38 mathematical papers out of Burkholder’s 53 listed (mathematical articles) and four comments by him as well as two commentaries on his work by his students … . This book is recommended for scientists in the field of mathematics and statistics.” (Adriana Horníková, Technometrics, Vol. 54 (1), February, 2012)


Table of contents (42 chapters)

  • Donald Burkholder’s Work in Martingales and Analysis

    Davis, Burgess (et al.)

    Pages 1-22

  • Don Burkholder’s work on Banach spaces

    Davis, Burgess (et al.)

    Pages 23-30

  • On a Class of Stochastic Approximation Processes

    Davis, Burgess (et al.)

    Pages 31-46

  • Sufficiency in the Undominated Case

    Davis, Burgess (et al.)

    Pages 47-56

  • Iterates of Conditional Expectation Operators

    Davis, Burgess (et al.)

    Pages 57-62

Buy this book

eBook $199.00
price for USA (gross)
  • ISBN 978-1-4419-7245-3
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $259.00
price for USA
  • ISBN 978-1-4419-7244-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Selected Works of Donald L. Burkholder
Editors
  • Burgess Davis
  • Renming Song
Series Title
Selected Works in Probability and Statistics
Copyright
2011
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media, LLC
eBook ISBN
978-1-4419-7245-3
DOI
10.1007/978-1-4419-7245-3
Hardcover ISBN
978-1-4419-7244-6
Series ISSN
2197-5825
Edition Number
1
Number of Pages
XXV, 729
Topics