Asymptotics for associated random variables. (English) Zbl 1249.62001
Berlin: Springer (ISBN 978-3-642-25531-1/hbk; 978-3-642-25532-8/ebook). x, 194 p. (2012).
The classical probability and statistics are based on the strong assumption of independent samples. However real phenomena are more complex and require to take into account some kind of dependence. Studies in this direction started about 1960 and nowadays there is a significant progress. The book under review presents in a unified way important results for associated random variables. Let us list the chapters:
1. Positive dependence. 2. Inequalities. 3. Almost sure convergence. 4. Convergence in distribution. 5. Convergence in distribution – functional results. Appendix A (General inequalities), Appendix B (General results on large deviations), Appendix C (Miscellaneous). References and Index.
The author provides a detailed analysis of different concepts of dependence and uses examples and counterexamples to illustrate notions and results. After this necessary preparation, he turns to the main goal, namely to study the asymptotic behavior of associated random variables. It is interesting to see fundamental laws such as the central limit theorem (CLT), the law of large numbers, large deviations, and invariance principles for sequences of associated random variables. Understandably, the rate of convergence is of a special concern. To give an example, the Berry-Esséen bound in the CLT for independent variables with finite third moment is of the form \(cn^{-1/2}\), while for associated variables the bound has the form \(cn^{-1/5}\). It is shown further how the limit theorems discussed in the book can be used when solving important statistical problems such as kernel estimation and regression for models with associated data.
The hook is well structured and carefully written. Most of the results are given with detailed proofs. There is a good balance between the author’s own results and results of other authors. It is clear that the progress achieved in this area requires new ideas and techniques which are more sophisticated than the machinery used for models with independence. It can be expected that the book will be met positively by researchers in modern stochastics.
1. Positive dependence. 2. Inequalities. 3. Almost sure convergence. 4. Convergence in distribution. 5. Convergence in distribution – functional results. Appendix A (General inequalities), Appendix B (General results on large deviations), Appendix C (Miscellaneous). References and Index.
The author provides a detailed analysis of different concepts of dependence and uses examples and counterexamples to illustrate notions and results. After this necessary preparation, he turns to the main goal, namely to study the asymptotic behavior of associated random variables. It is interesting to see fundamental laws such as the central limit theorem (CLT), the law of large numbers, large deviations, and invariance principles for sequences of associated random variables. Understandably, the rate of convergence is of a special concern. To give an example, the Berry-Esséen bound in the CLT for independent variables with finite third moment is of the form \(cn^{-1/2}\), while for associated variables the bound has the form \(cn^{-1/5}\). It is shown further how the limit theorems discussed in the book can be used when solving important statistical problems such as kernel estimation and regression for models with associated data.
The hook is well structured and carefully written. Most of the results are given with detailed proofs. There is a good balance between the author’s own results and results of other authors. It is clear that the progress achieved in this area requires new ideas and techniques which are more sophisticated than the machinery used for models with independence. It can be expected that the book will be met positively by researchers in modern stochastics.
Reviewer: Jordan M. Stoyanov (Newcastle upon Tyne)
MSC:
| 62E20 | Asymptotic distribution theory in statistics |
| 62-02 | Research exposition (monographs, survey articles) pertaining to statistics |
| 60E05 | Probability distributions: general theory |
| 62H20 | Measures of association (correlation, canonical correlation, etc.) |
| 60F17 | Functional limit theorems; invariance principles |
| 60Fxx | Limit theorems in probability theory |
| 60F05 | Central limit and other weak theorems |
| 60F10 | Large deviations |
