Mixing is concerned with the analysis of dependence between sigma-fields defined on the same underlying probability space. It provides an important tool of analysis for random fields, Markov processes, central limit theorems as well as being a topic of current research interest in its own right. The aim of this monograph is to provide a study of applications of dependence in probability and statistics. It is divided in two parts, the first covering the definitions and probabilistic properties of mixing theory. The second part describes mixing properties of classical processes and random fields as well as providing a detailed study of linear and Gaussian fields. Consequently, this book will provide statisticians dealing with problems involving weak dependence properties with a powerful tool.
Inhalt
1. General properties.- 1.1. Dependence of ?-fields.- 1.2. Basic tools.- 1.2.1. Reconstruction techniques.- 1.2.2. Covariance inequalities.- 1.3. Mixing.- 1.3.1. Mixing random fields.- 1.3.2. Mixing processes.- 1.3.3. Weak conditions for processes.- 1.3.4. Miscellany.- 1.4. Tools.- 1.4.1. Moment inequalities.- 1.4.2. Exponential inequalities.- 1.4.3. Maximal inequalities.- 1.5. Central limit theorem.- 1.5.1 Sufficient conditions.- 1.5.2. Convergence rates.- 1.5.3. Dimension dependent rates.- 2. Examples.- 2.1. Gaussian random fields.- 2.1.1. An explicit bound.- 2.1.2. Mixing rates.- 2.2. Gibbs fields.- 2.2.1 Dobrushin theory.- 2.2.1.1. Comparison between specifications.- 2.2.1.2 Dobrushin's condition.- 2.2.1.3. Mixing condition.- 2.2.2 Markov fields.- 2.2.2.1. Potentials.- 2.2.3 Non compact case.- 2.2.3.1. Point processes.- 2.2.3.2. Diffusions.- 2.3. Linear fields.- 2.3.1. Independent innovations.- 2.3.2. Dependent innovations.- 2.3.3. Proofs.- 2.3.4. Miscellany.- 2.4. Markov processes.- 2.4.0.1. A class of non linear models.- 2.4.0.2. Dynamical systems approach.- 2.4.0.3. Annealing.- 2.4.1. Polynomial AR processes.- 2.4.1.1. Bilinear models.- 2.4.1.2. ARMA models.- 2.4.2. Nonlinear processes.- 2.4.2.1. ARX(k, q) nonlinear processes.- 2.4.2.2. AR(1) nonlinear processes.- 2.4.2.3. Financial nonlinear processes.- 2.5. Continuous time processes.- 2.5.1. Markov processes.- 2.5.2. Operators.- 2.5.3. Diffusion processes.- 2.5.4. Hypermixing.- 2.5.5. Hypercontractivity.- 2.5.6. Ultracontractivity.- 2.5.7. General SDE.