Introduction to Strong Mixing Conditions Volumes 1,2 and 3

Richard C. Bradley

Kendrick Press, 2007. Paperback ISBN 0-9740427-6-5 (vol.1), 0-9740427-7-3 (vol.2), 0-9740427-8-1 (vol.3), 0-9740427-9-X (series). Hardback ISBN 0-9793183-1-9 (vol. 1), 0-9793183-2-7 (vol. 2), 0-9793183-3-5 (vol. 3), 0-9793183-4-3 (series). Volume 1: xviii + 539 pp. Volume 2: xii + 553 pp. Volume 3: xii + 597 pp. Price for each volume is $70 (paperback), $105 (hardback).

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.

Chapter headings:

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

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

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

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

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