Martingale Limit Theory and Its Application (Probability and Mathematical Statistics)

Title page......Page 1
Contents......Page 3
Preface......Page 7
1.2. Historical Interlude......Page 10
1.3. The Martingale Convergence Theorem......Page 11
1.4. Comments on Classical Limit Theory and Its Analogs......Page 12
1.5. On the Repertoire of Available Limit Theory......Page 15
1.6. Martingale Limit Theorems Generalizing Those for Sums of Independent Random Variables......Page 17
1.7. Martingale Limit Theorems Viewed as Rate of Convergence. Results in the Martingale Convergence Theorem......Page 19
2.1. Introduction......Page 22
2.2. Basic Inequalities......Page 23
2.3. The Martingale Convergence Theorem......Page 26
2.4. Square Function Inequalities......Page 32
2.5. Weak Law of Large Numbers......Page 38
2.6. Strong Law of Large Numbers......Page 40
2.7. Convergence in L^p......Page 50
3.1. Introduction......Page 60
3.2. The Central Limit Theorem......Page 61
3.3. Toward a General Central Limit Theorem......Page 74
3.4. Raikov-Type Results in the Martingale CLT......Page 78
3.5. Reverse Martingales and Martingale Tail Sums......Page 85
3.6. Rates of Convergence in the C T......Page 90
4.2. Invariance Principles in CLT......Page 106
4.3. Rates of Convergence for the Invariance Principle in the CLT......Page 118
4.4. The Law of the Iterated Logarithm and Its Invariance Principle......Page 124
5.1. Introduction......Page 136
5.3. The Central Limit Theorem......Page 137
5.4. Functional Forms of the Central Limit Theorem and Law of the Iterated Logarithm......Page 149
5.5. The Central Limit Theorem via Approximation to the Distribution of the Stationary Process......Page 156
6.1. Introduction......Page 164
6.2. Asymptotic Behaviour of the Maximum Likelihood Estimator......Page 165
6.3. Conditional Least Squares......Page 181
6.4. Quadratic Functions of Discrete Time Series......Page 191
6.5. The Method of Moments......Page 204
7.1. Exchangeable Sequences......Page 210
7.2. Limit Laws for Subsequences of Sequences of Random Variables......Page 214
7.3. Limit Laws for Subadditive Processes......Page 222
7.4. The Hawkins Random Sieve......Page 235
7.5. Genetic Balance When the Population Size Is Varying......Page 244
7.6. Stochastic Approximation......Page 247
7.7. On Adaptive Control of Linear Systems......Page 261
I. The Skorokhod Representation......Page 278
II. Weak Convergence on Function Spaces......Page 282
III. Mixing Inequalities......Page 285
IV. Stationarity and Ergodicity......Page 289
V. Miscellanea......Page 290
References......Page 294
Index to Theorems, Corollaries, and Examples......Page 308
Index......Page 310