Weak Dependence With Examples and Applications

by Dedecker, Jerome; Doukhan, Paul; Lang, Gabriel; Leon, Jose Rafael; Louhichi, Sana

• ISBN13: 9780387699516
• ISBN10: 0387699511
• Format: Paperback
• Copyright: 2007-07-06
• Publisher: Springer Verlag

Summary

This monograph is aimed at developing Doukhan/Louhichi's (1999) idea to measure asymptotic independence of a random process. The authors propose various examples of models fitting such conditions such as stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory. Most of the commonly used stationary models fit their conditions. The simplicity of the conditions is also their strength. The main existing tools for an asymptotic theory are developed under weak dependence. They apply the theory to nonparametric statistics, spectral analysis, econometrics, and resampling. The level of generality makes those techniques quite robust with respect to the model. The limit theorems are sometimes sharp and always simple to apply. The theory (with proofs) is developed and the authors propose to fix the notation for future applications. A large number of research papers deals with the present ideas; the authors as well as numerous other investigators participated actively in the development of this theory. Several applications are still needed to develop a method of analysis for (nonlinear) times series and they provide here a strong basis for such studies.

Table of Contents

Preface	p. v
List of notations	p. xiii
Introduction	p. 1
From independence to dependence	p. 1
Mixing	p. 4
Mixingales and near epoch dependence	p. 5
Association	p. 6
Nonmixing models	p. 8
Weak dependence	p. 9
Function spaces	p. 10
Weak dependence	p. 11
¿, ¿, ¿ and (¿-coefficients	p. 12
¿ and ¿-coefficients	p. 14
<$>\tilde \alpha<$>, <$>\tilde \beta<$> and <$>\tilde \phi<$>-coefficients	p. 16
Projective measure of dependence	p. 19
Models	p. 21
Bernoulli shifts	p. 21
Volterra processes	p. 22
Noncausal shifts with independent inputs	p. 24
Noncausal shifts with dependent inputs	p. 25
Causal shifts with independent inputs	p. 31
Causal shifts with dependent inputs	p. 32
Markov sequences	p. 33
Contracting Markov chain	p. 35
Nonlinear AR(d) models	p. 36
ARCH-type processes	p. 36
Branching type models	p. 37
Dynamical systems	p. 38
Vector valued LARCH(∞) processes	p. 42
Chaotic expansion of LARCH(∞) models	p. 43
Bilinear models	p. 48
¿-dependent models	p. 53
Associated processes	p. 53
Gaussian processes	p. 55
Interacting particle systems	p. 56
Other models	p. 59
Random AR models	p. 59
Integer valued models	p. 61
Random fields	p. 62
Continuous time	p. 65
Tools for non causal cases	p. 67
Indicators of weakly dependent processes	p. 67
Low order moments inequalities	p. 69
Variances	p. 69
A (2 + ¿)-order momentbound	p. 70
Combinatorial moment inequalities	p. 73
Marcinkiewicz-Zygmundtype inequalities	p. 77
Rosenthal type inequalities	p. 79
A first exponential inequality	p. 82
Cumulants	p. 84
General properties of cumulants	p. 84
A second exponential inequality	p. 93
From weak dependence to the exponential bound	p. 96
Tightness criteria	p. 98
Tools for causal cases	p. 103
Comparison results	p. 103
Covariance inequalities	p. 110
A covariance inequality for ¿1	p. 110
A covariance inequality for <$>\tilde \beta<$> and <$>\tilde \phi<$>	p. 111
Coupling	p. 114
A coupling result for real valued random variables	p. 115
Coupling in higher dimension	p. 116
Exponential and moment inequalities	p. 119
Bennett-type inequality	p. 120
Burkholder's inequalities	p. 123
Rosenthal inequalities using Rio techniques	p. 125
Rosenthal inequalities for ¿1-dependent sequences	p. 130
Rosenthal inequalities under projective conditions	p. 131
Maximal inequalities	p. 132
Applications of SLLN	p. 135
Stochastic algorithms with non causal dependent input	p. 135
Weakly dependent noise	p. 137
¿1-dependent noise	p. 140
Examples of application	p. 142
Robbins-Monro algorithm	p. 142
Kiefer-Wolfowitz algorithm	p. 143
Weighted dependent triangular arrays	p. 143
Linear regression	p. 145
Central limit theorem	p. 153
Non causal case: stationary sequences	p. 153
Lindeberg method	p. 155
Proof of the main results	p. 158
Rates of convergence	p. 161
Non causal random fields	p. 163
Conditional central limit theorem (causal)	p. 173
Definitions and preliminary lemmas	p. 174
Invariance of the conditional variance	p. 176
End of the proof	p. 178
Applications	p. 182
Stable convergence	p. 182
Sufficient conditions for stationary sequences	p. 184
¿-dependent sequences	p. 189
<$>\tilde \alpha<$> and <$>\tilde \phi<$>-dependent sequences	p. 192
Sufficient conditions for triangular arrays	p. 194
Donsker principles	p. 199
Non causal stationary sequences	p. 199
Non causal random fields	p. 200
Moment inequality	p. 201
Finite dimensional convergence	p. 202
Tightness	p. 205
Conditional (causal) invariance principle	p. 205
Preliminaries	p. 206
Finite dimensional convergence	p. 207
Relative compactness	p. 208
Applications	p. 209
Sufficient conditions for stationary sequences	p. 209
Sufficient conditions for triangular arrays	p. 212
Law of the iterated logarithm (LIL)	p. 213
Bounded LIL under a non causal condition	p. 213
Causal strong invariance principle	p. 214
The empirical process	p. 223
A simple condition for the tightness	p. 224
¿-dependent sequences	p. 225
<$>\tilde \alpha<$>, <$>\tilde \beta<$> and <$>\tilde \phi<$>-dependent sequences	p. 231
¿ and ¿-dependent sequences	p. 233
Empirical copula processes	p. 234
Random fields	p. 236
Functional estimation	p. 247
Some non-parametric problems	p. 247
Kernel regression estimates	p. 248
Second order and CLT results	p. 249
Almost sure convergence properties	p. 252
MISE for <$>\tilde \beta<$>-dependent sequences	p. 254
General kernels	p. 260
Spectral estimation	p. 265
Spectral densities	p. 265
Periodogram	p. 269
Whittle estimation	p. 274
Spectral density estimation	p. 275
Second order estimate	p. 277
Dependence coefficients	p. 279
Econometric applications and resampling	p. 283
Econometrics	p. 283
Unit root tests	p. 284
Parametric problems	p. 285
A semi-parametric estimation problem	p. 285
Bootstrap	p. 287
Block bootstrap	p. 288
Bootstrapping GMM estimators	p. 288
Conditional bootstrap	p. 290
Sieve bootstrap	p. 290
Limit variance estimates	p. 292
Moments, cumulants and weak dependence	p. 293
Estimation of the limit variance	p. 295
Law of the large numbers	p. 297
Central limit theorem	p. 299
A non centered variant	p. 302
Bibliography	p. 305
Index	p. 317
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