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.