| 目录 |
| | Preface | v |
| 1. | Introduction | 1 |
| 2. | Convergence to local time | 5 |
| 2.1. | Local time: definition and existence | 6 |
| 2.1.1. | Local time of Gaussian process | 8 |
| 2.1.2. | Local time of Levy process | 11 |
| 2.1.3. | Local time of semimartingale | 12 |
| 2.2. | Convergence to integral functionals of stochastic processes | 17 |
| 2.2.1. | Existence of integral10g(Xs)ds | 18 |
| 2.2.2. | Convergence to integral10g(Xs)ds | 21 |
| 2.2.3. | Supplement and generalization | 23 |
| 2.3. | Convergence to local time | 27 |
| 2.3.1. | Strong smooth array: definition and examples | 28 |
| 2.3.2. | Convergence to local time: framework I | 32 |
| 2.3.3. | Example: linear processes | 38 |
| 2.3.4. | Proofs of Corollaries 2.3 and 2.4 | 41 |
| 2.3.5. | Convergence to local time: framework II | 47 |
| 2.3.6. | Example: stationary processes | 55 |
| 2.4. | Convergence to self-intersection local time | 60 |
| 2.5. | Uniform approximation to local time | 63 |
| 2.5.1. | An invariance principle | 64 |
| 2.5.2. | Uniform approximation to local time | 67 |
| 2.5.3. | Uniform approximation to local time: random bandwidth | 68 |
| 2.5.4. | Example: linear processes | 71 |
| 2.5.5. | Proofs of main results | 72 |
| 2.6. | Bibliographical Notes | 78 |
| 3. | Convergence to a mixture of normal distributions | 81 |
| 3.1. | Convergence on product space | 81 |
| 3.2. | Convergence to a process with stationary independent increments | 85 |
| 3.2.1. | Central limit theorem for dependent random variables | 86 |
| 3.2.2. | Functional central limit theorems for martingales | 90 |
| 3.2.3. | Multivariate martingale limit theorem | 92 |
| 3.2.4. | Convergence to a stable Levy process | 94 |
| 3.3. | Convergence to a mixture of normal distributions: martingale arrays | 96 |
| 3.3.1. | A framework | 96 |
| 3.3.2. | Limit theorem for martingale: under conditional variance condition (CVC) | 97 |
| 3.3.3. | Examples and remarks | 101 |
| 3.4. | Martingale limit theorem revisited | 103 |
| 3.4.1. | Limit theorem for martingale: under convergence in distribution for conditional variance | 104 |
| 3.4.2. | Two examples | 106 |
| 3.4.3. | Proof of Theorem 3.13 | 111 |
| 3.5. | Convergence to a mixture of normal distributions: beyond martingale arrays | 119 |
| 3.5.1. | Proof of Theorem 3.16 | 123 |
| 3.5.2. | Some subsidiary propositions | 125 |
| 3.6. | Convergence to a mixture of normal distributions: zero energy functionals | 130 |
| 3.7. | Uniform convergence for a class of martingales | 132 |
| 3.7.1. | A framework and applications | 132 |
| 3.7.2. | Remarks and extension | 137 |
| 3.7.3. | Examples: Identification of (3.125) | 140 |
| 3.8. | Limit theorems for continuous time martingales | 145 |
| 3.8.1. | Convergence to a continuous Gaussian process | 145 |
| 3.8.2. | Convergence to a mixture of Gaussian processes | 146 |
| 3.9. | Bibliographical Notes | 148 |
| 4. | Convergence to stochastic integrals | 151 |
| 4.1. | Definition of stochastic integrals | 151 |
| 4.1.1. | Stochastic integrals with respect to a Brownian motion | 152 |
| 4.1.2. | Stochastic integrals with respect to a (local) square integrable martingale | 154 |
| 4.1.3. | Stochastic integrals with respect to a local (semi-) martingale | 155 |
| 4.1.4. | Multivariate stochastic integrals | 156 |
| 4.1.5. | Properties of stochastic integrals | 157 |
| 4.1.6. | Martingale representation theorem and It©þ's formula | 158 |
| 4.2. | Weak convergence of stochastic integrals | 159 |
| 4.3. | Weak convergence of stochastic integrals: multivariate extension | 161 |
| 4.4. | Convergence to stochastic integrals: random arrays | 162 |
| 4.5. | Convergence to stochastic integrals: beyond the semi-martingale | 165 |
| 4.5.1. | LPWW decomposition theorem | 166 |
| 4.5.2. | Long memory processes | 170 |
| 4.5.3. | Short memory processes | 174 |
| 4.5.4. | LPWW decomposition theorem: multivariate extension | 179 |
| 4.5.5. | Extension to a-mixing sequences | 181 |
| 4.6. | Bibliographical Notes | 186 |
| 5. | Nonlinear cointegrating regression | 189 |
| 5.1. | Nonparametric estimation | 189 |
| 5.1.1. | Nadaraya-Watson estimator | 190 |
| 5.1.2. | Nadaraya-Watson estimator: certain extensions | 195 |
| 5.1.3. | Bias analysis and local linear estimator | 199 |
| 5.1.4. | Uniform convergence for local linear estimator | 202 |
| 5.1.5. | Multivariate contingrating regression | 207 |
| 5.2. | Parametric estimation | 210 |
| 5.2.1. | Weak consistency | 210 |
| 5.2.2. | Asymptotic distribution | 215 |
| 5.3. | Model specification testing | 219 |
| 5.4. | Bibliographical Notes | 225 |
| | Appendix A Concepts of stochastic processes | 229 |
| | Appendix B Metric space | 237 |
| | Appendix C Convergence of probability measure | 241 |
| | Bibliography | 245 |
| | Index | 257 |
