Mixing： Properties and Examples

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mixing不知如何翻译成中文。数据流是不稳定的。一种分析方法是假定随机变量在一段时间后的平均值近似稳定。目前也是机器学习的一个方法。
Paul Doukhan
Introduction................................................. ..... .. vii
Notations .................... . ....... X1

Generalproperties.so......o.......n...........-o......--......o.....1

Dependence of (it-fields ...............................................................3
B asic tools ................................... . ..........................................7
1.2.1. Reconstruction techniques ..................................................7
1 .2.2. C0variance inequalities ........................................ . ............9
Mixing ................................................................................ 15
1.3. l. Mixing random fields ..................................................... 1 6
1 .3.2 Mixing processes .......................................................... 18
1.3.3. Weak conditions for processes .......................................... 2 1
l .3.4 Miscellany .................................................................. 2 2
Tools ..................................................................... . ............ 2 5
1.4 1 Moment inequalities ....................................................... 25
1.4.2. Exponential inequalities ................................................... 3 3
1.4 3 Maximal inequalities ...................................................... 4 0
Central limit theorem ................................................................ 4 s
1.5 l Sufficient conditions ...................................................... 45
1 .5.2. Convergence rates ......................................................... 4 8
1.5 3 Dimension dependent rates ................................ . .............. 4 9
Examples ............................... 55
Gaussian random fields ............................................................. 57
2.1.1. An explicit bound .......................................................... 58
2. 1.2. Mixing rates ................................................................ 6 1
Gibbs fields .......................................................................... 63
2.2. 1 Dobrushin theory .......................................................... 63
2.2.1.1. Comparison between specifications.. 64
2.2.1.2. Dobrushin's c0nditinn 65
2.2.1.3. Mixing condition 66
2.2.2 Markov fields .............................................................. 6 7
2.2.2.1. Potentials 67
2.2.3 Non compact case ......................................................... 7 0
2.2.3.1. Point processes 7 1
2.2.3.2. Diffusions . 7 2

2.3. Linear fields ........................ _.................................................. 7 5
2.3 1 Independent innovations .................................................. 7 7
2.3.2. Dependent innovatlons .................................................... 80
2.3 3 Proofs ....................................................................... 8 1
2.3.4. Miscellany ................................................................. 84
2.4. Markov processes ................................................................... 8 7
2.4.0.1. A class of non linear models 9 3
2.4.0.2. Dynamical systems approach 9 3
2.4.0.3. Annealing 94
2.4.1. Polynomial AR processes ................................................ 9 6
2.4.1.1. Bilinear models 9 8
2.4.1.2. ARMA mndels 98
2.4.2. Nonlinear processes ...................................................... 100
2.4.2.1. ARX(k, q) nonlinear processes 1 00
2.4.2.2. AR(1) nonlinear processes 1 04
2.4.2.3. Financial nonlinear processes 106
2.5. Continuous time processes ........................................................ 1 1 1
2.5. l . Mark0v processes ........................................................1 1 1
2.5.2 Operators ..................................................................111
2.5 .3 Diffusion processes ....................................... . .............. 113
2.5.4 Hypermixing....... .......................................................118
2.5.5 Hypercontractiv1ty ........................................................1 19
2.5.6 Ulnacontractivity ......................................................... 1 2 1
2.5.7 General SDE .............................................................. 122
Bibliography ................. . .................... . .................... ....125
Index ......................... . ........... ........... ....... 141

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