Limit Theory for Mixing Dependent Random Variables

by Lin, Cheng-Yen; Chuanrong, Lu; Zhengyan, Lin; Lu, Chuanrong

• ISBN13: 9780792342199
• ISBN10: 0792342194
• Format: Hardcover
• Copyright: 1997-08-01
• Publisher: Kluwer Academic Pub

Summary

For many practical problems, observations are not independent. In this book, limit behaviour of an important kind of dependent random variables, the so-called mixing random variables, is studied. Many profound results are given, which cover recent developments in this subject, such as basic properties of mixing variables, powerful probability and moment inequalities, weak convergence and strong convergence (approximation), limit behaviour of some statistics with a mixing sample, and many useful tools are provided. Audience: This volume will be of interest to researchers and graduate students in the field of probability and statistics, whose work involves dependent data (variables).

Table of Contents

Preface	vii
Part I Introduction	1	(38)
Chapter 1 Definitions and Basic Inequalities	3	(12)
1.1 Definitions	3	(4)
1.2 Basic inequalities	7	(8)
Chapter 2 Moment Estimations of Partial Sums	15	(24)
2.1 Variances of Partial sums	15	(9)
2.2 Further inequalitis	24	(15)
Part II Weak Convergence	39	(150)
Chapter 3 Weak Convergence for XXX-mixing Sequences	41	(30)
3.1 Necessary and sufficient conditions for the CLT	41	(10)
3.2 Sufficient conditions for CLT and WIP	51	(12)
3.3 The CLT and WIP when the variance is infinite	63	(8)
Chapter 4 Weak Convergence for XXX-mixing Sequences	71	(52)
4.1 The WIP when the moments of order 2 are finite	73	(5)
4.2 The WIP when moments of higher than two orders	78	(10)
4.3 A generalized result when moments of higher than two orders	88	(16)
4.4 The WIP when the variance is infinite	104	(19)
Chapter 5 Weak Convergence for XXX-mixing Sequences	123	(18)
5.1 The WIP when the moments of order 2 are finite	124	(12)
5.2 The Ibragimov-Linnik-Iosifescu conjecture	136	(5)
Chapter 6 Weak Convergence for Mixing Random Fields	141	(28)
6.1 The CLT for mixing random fields	141	(8)
6.2 Convergence of finite dimensional distributions	149	(11)
6.3 Tightness	160	(9)
Chapter 7 The Berry-Esseen Inequality and the Rate of Weak Convergence	169	(20)
7.1 Rate of convergence in distribution for XXX-mixing and XXX-mixing sequences	169	(12)
7.2 The rate of weak convergence for a XXX-mixing sequence	181	(8)
Part III Almost Sure Convergence and Strong Approximations	189	(120)
Chapter 8 Laws of Large Numbers and Complete Convergence	191	(50)
8.1 Weak law of large numbers	191	(8)
8.2 Strong laws of large numbers	199	(2)
8.3 Complete convergence for XXX-mixing sequences	201	(7)
8.4 Complete convergence for XXX-mixing sequences	208	(14)
8.5 Complete convergence for XXX-mixing sequences	222	(9)
8.6 A further discussion on the complete convergence	231	(10)
Chapter 9 Strong Approximations	241	(28)
9.1 Strong approximations for a XXX-mixing sequence	241	(6)
9.2 Strong approximations for a XXX-mixing sequence	247	(16)
9.3 Strong approximations for a XXX-mixing sequence	263	(6)
Chapter 10 The Increments of Partial Sums	269	(18)
10.1 Some lemmas	269	(8)
10.2 How big are the increments when the moment generation functions exist?	277	(6)
10.3 How big are the increments when the moment generating functions do not exist?	283	(4)
Chapter 11 Strong Approximations for Mixing Random Fields	287	(22)
11.1 Strong approximations of a XXX-mixing random field	288	(13)
11.2 Strong approximations of a XXX-mixing random fields	301	(8)
Part IV Statistics of a Dependent Sample	309	(97)
Chapter 12 Empirical Processes	311	(36)
12.1 Weak convergence	312	(5)
12.2 Weighted weak convergence	317	(14)
12.3 Strong approximations	331	(10)
12.4 Moduli of continuity of empirical processes	341	(6)
Chapter 13 Convergence of Some Statistics with a Mixing Sample	347	(32)
13.1 U-Statistics	347	(13)
13.2 Error variance estimations in linear models	360	(9)
13.3 Density estimations	369	(10)
Chapter 14 Strong Approximations for Other Kinds of Dependent Random Variables	379	(27)
14.1 Lacunary trigonometric series with weights	379	(13)
14.2 A class of Gaussian sequences	392	(5)
14.3 The non-negative additive functional of a Markov process	397	(9)
Appendix	406	(3)
References	409	(16)
Index	425