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Richard C. Bradley

Introduction to Strong Mixing Conditions
Volume 1

Author:Richard C. Bradley

Richard Bradley is Professor of Mathematics at Indiana University. He is an active researcher in the field treated in this series.

Kendrick Press, 2007.
Volume 1: xviii + 539 pp.
Price for each volume is $80 (paperback), $115 (hardback).
Paperback ISBN 0-9740427-6-5 (vol.1), 0-9740427-9-X (series)
Hardback ISBN 0-9793183-1-9 (vol. 1), 0-9793183-4-3 (series).

For many phenomena of the real world, observations in the past and present may have considerable influence on observations in the near future, but rather weak influence on observations in the far future. Random sequences that satisfy "strong mixing conditions" are used to model such phenomena. This three-volume series is an introduction to the theory of strong mixing conditions. All three volumes deal primarily with (1) the central limit theorem under various strong mixing conditions and (2) basic structural properties of strong mixing conditions. Well-known constructions from the literature are used to illustrate various subtleties and limitations in connection with both the central limit theory and the structural properties involving such conditions. The proofs are given in much more detail than in most papers and monographs, in order to help newcomers to the field. The main prerequisite for the study of these volumes is a graduate-level command of real analysis and measure-theoretic probability theory.

Volume 1

  • 1 - Introduction to the (Rosenblatt) strong mixing condition
  • 2 - Connections with ergodic theory
  • 3 - Five classic strong mixing conditions
  • 4 - Norms and connections with interpolation theory
  • 5 - Some other strong mixing conditions
  • 6 - Independent pairs of 6-fields
  • 7 - Markov chains
  • 8 - Second order properties
  • 9 - Stationary Gaussian sequences
  • 10 - Central limit theorems under the strong mixing condition
  • 11 - Central limit theorems under P-mixing, P*-mixing and related conditions
  • 12 - General limiting behavior of partial sums under strong mixing
  • 13 - A brief review of some other topics

 
Richard C. Bradley

Volume 2

Volume 2: xii + 553 pp.
Price for each volume is $80 (paperback), $115 (hardback)
Paperback ISBN 0-9740427-7-3 (vol.2), 0-9740427-9-X (series)
Hardback ISBN 0-9793183-2-7 (vol. 2), 0-9793183-4-3 (series)





Volume 2

  • 14 - Relevant material (mostly) from Volume 1
  • 15 - Direct approximation by martingale differences, a` la Gordin
  • 16 - Direct approximation by independent random variables, a` la Berkes and Philipp
  • 17 - Central limit theorems under "minimal" conditions
  • 18 - A two-part mixing assumption
  • 19 - Tightness, shift-tightness and complete dissipation under strong mixing
  • 20 - Periodicity and related topics for non-Markovian strictly stationary sequences
  • 21 - Markov chains (revisited)
  • 22 - Dichotomies for some dependence coefficients
  • 23 - Linear dependence conditions (again)
  • 24 - Some other dependence conditions

 
Richard C. Bradley

Volume 3

Volume 3: xii + 597 pp.
Price for each volume is $80 (paperback), $115 (hardback).
Paperback ISBN 0-9740427-8-1 (vol.3), 0-9740427-9-X (series).
Hardback ISBN 0-9793183-3-5 (vol. 3), 0-9793183-4-3 (series).





Volume 3

  • 25 - Relevant material from Volumes 1 and 2
  • 26 - Examples involving prescribed mixing rates
  • 27 - Stationary Gaussian processes (revisited)
  • 28 - Random fields I: Linear dependence conditions and spectral density
  • 29 - Random fields II: Strong mixing conditions
  • 30 - Counterexamples to the central limit theorem: Markov chains, mixing rates a` la Davydov
  • 31 - Counterexamples with arbitrarily fast mixing rates
  • 32 - Some miscellaneous counterexamples
  • 33 - Counterexamples involving quantiles
  • 34 - P-mixing counterexamples

 

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