| Introduction | p. 1 |
| Independent Random Variables | |
| Classical Limit Theorems, Inequalities and Other Tools | p. 7 |
| Classical Limit Theorems | p. 7 |
| The Weak Law, Strong Law and Law of the Iterated Logarithm | p. 8 |
| The Central Limit Theorem | p. 9 |
| Cramér's Moderate Deviation Theorem | p. 11 |
| Exponential Inequalities for Sample Sums | p. 11 |
| Self-Normalized Sums | p. 11 |
| Tail Probabilities for Partial Sums | p. 13 |
| Characteristic Functions and Expansions Related to the CLT | p. 17 |
| Continuity Theorem and Weak Convergence | p. 18 |
| Smoothing, Local Limit Theorems and Expansions | p. 19 |
| Supplementary Results and Problems | p. 21 |
| Self-Normalized Large Deviations | p. 25 |
| A Classical Large Deviation Theorem for Sample Sums | p. 25 |
| A Large Deviation Theorem for Self-Normalized Sums | p. 27 |
| Representation by Supremum over Linear Functions of (Sn, Vn2) | p. 27 |
| Proof of Theorem 3.1 | p. 28 |
| Supplementary Results and Problems | p. 31 |
| Weak Convergence of Self-Normalized Sums | p. 33 |
| Self-Normalized Central Limit Theorem | p. 33 |
| Non-Normal Limiting Distributions for Self-Normalized Sums | p. 37 |
| Supplementary Results and Problems | p. 38 |
| Stein's Method and Self-Normalized Berry-Esseen Inequality | p. 41 |
| Stein's Method | p. 41 |
| The Stein Equation | p. 41 |
| Stein's Method: Illustration of Main Ideas | p. 44 |
| Normal Approximation for Smooth Functions | p. 46 |
| Concentration Inequality and Classical Berry-Esseen Bound | p. 49 |
| A Self-Normalized Berry-Esseen Inequality | p. 52 |
| Proof: Outline of Main Ideas | p. 53 |
| Proof: Details | p. 55 |
| Supplementary Results and Problems | p. 60 |
| Self-Normalized Moderate Deviations and Laws of the Iterated Logarithm | p. 63 |
| Self-Normalized Moderate Deviations: Normal Case | p. 63 |
| Proof of the Upper Bound | p. 64 |
| Proof of the Lower Bound | p. 66 |
| Self-Normalized Moderate Deviations: Stable Case | p. 69 |
| Preliminary Lemmas | p. 70 |
| Proof of Theorem 6.6 | p. 76 |
| Self-Normalized Laws of the Iterated Logarithm | p. 81 |
| Supplementary Results and Problems | p. 84 |
| Cramer-Type Moderate Deviations for Self-Normalized Sums | p. 87 |
| Self-Normalized Cramer-Type Moderate Deviations | p. 87 |
| Proof of Theorems | p. 90 |
| Proof of Theorems 7.2, 7.4 and Corollaries | p. 90 |
| Proof of Theorem 7.1 | p. 91 |
| Proof of Propositions | p. 94 |
| Application to Self-Normalized LIL | p. 96 |
| Cramer-Type Moderate Deviations for Two-Sample t-Statistics | p. 104 |
| Supplementary Results and Problems | p. 106 |
| Self-Normalized Empirical Processes and U-Statistics | p. 107 |
| Self-Normalized Empirical Processes | p. 107 |
| Self-Normalized U-Statistics | p. 108 |
| The Hoeffding Decomposition and Central Limit Theorem | p. 109 |
| Self-Normalized U-Statistics and Berry-Esseen Bounds | p. 109 |
| Moderate Deviations for Self-Normalized U-Statistics | p. 110 |
| Proofs of Theorems 8.5 and 8.6 | p. 111 |
| Main Ideas of the Proof | p. 111 |
| Proof of Theorem 8.6 | p. 112 |
| Proof of Theorem 8.5 | p. 113 |
| Proof of Proposition 8.7 | p. 113 |
| Supplementary Results and Problems | p. 119 |
| Martingales and Dependent Random Vectors | |
| Martingale Inequalities and Related Tools | p. 123 |
| Basic Martingale Theory | p. 123 |
| Conditional Expectations and Martingales | p. 123 |
| Martingale Convergence and Inequalities | p. 125 |
| Tangent Sequences and Decoupling Inequalities | p. 125 |
| Construction of Decoupled Tangent Sequences | p. 126 |
| Exponential Decoupling Inequalities | p. 126 |
| Exponential Inequalities for Martingales | p. 128 |
| Exponential Inequalities via Decoupling | p. 128 |
| Conditionally Symmetric Random Variables | p. 132 |
| Exponential Supermartingales and Associated Inequalities | p. 134 |
| Supplementary Results and Problems | p. 135 |
| A General Framework for Self-Normalization | p. 137 |
| An Exponential Family of Supermartingales Associated with Self-Normalization | p. 137 |
| The I.I.D. Case and Another Derivation of (3.8) | p. 137 |
| A Representation of Self-Normalized Processes and Associated Exponential Supermartingales | p. 138 |
| Canonical Assumptions and Related Stochastic Models | p. 139 |
| Continuous-Time Martingale Theory | p. 140 |
| Doob-Meyer Decomposition and Locally Square-Integrable Martingales | p. 141 |
| Inequalities and Stochastic Integrals | p. 143 |
| Supplementary Results and Problems | p. 146 |
| Pseudo-Maximization via Method of Mixtures | p. 149 |
| Pseudo-Maximization and Laplace's Method | p. 149 |
| A Class of Mixing Densities | p. 150 |
| Application of Method of Mixtures to Boundary Crossing Probabilities | p. 152 |
| The Robbins-Siegmund Boundaries for Brownian Motion | p. 152 |
| Extensions to General Self-Normalized Processes | p. 154 |
| Supplementary Results and Problems | p. 157 |
| Moment and Exponential Inequalities for Self-Normalized Processes | p. 161 |
| Inequalities of Caballero, Fernandez and Nualart, Graversen and Peskir, and Kikuchi | p. 161 |
| Moment Bounds via the Method of Mixtures | p. 164 |
| Gaussian Mixing Densities | p. 165 |
| The Mixing Density Functions in Sect. 11.2 | p. 167 |
| Applications and Examples | p. 174 |
| Proof of Lemma 8.11 | p. 174 |
| Generalizations of Theorems 12.1, 12.2 and 12.3 | p. 175 |
| Moment Inequalities Under Canonical Assumption for a Restricted Range | p. 176 |
| Supplementary Results and Problems | p. 177 |
| Laws of the Iterated Logarithm for Self-Normalized Processes | p. 179 |
| Stout's LIL for Self-Normalized Martingales | p. 179 |
| A Universal Upper LIL | p. 182 |
| Compact LIL for Self-Normalized Martingales | p. 186 |
| Supplementary Results and Problems | p. 190 |
| Multivariate Self-Normalized Processes with Matrix Normalization | p. 193 |
| Multivariate Extension of Canonical Assumptions | p. 193 |
| Matrix Sequence Roots for Self-Normalization | p. 193 |
| Canonical Assumptions for Matrix-Normalized Processes | p. 194 |
| Moment and Exponential Inequalities via Pseudo-Maximization | p. 196 |
| LIL and Boundary Crossing Probabilities for Multivariate Self-Normalized Processes | p. 201 |
| Supplementary Results and Problems | p. 202 |
| Statistical Applications | |
| The t-Statistic and Studentized Statistics | p. 207 |
| Distribution Theory of Student's t-Statistics | p. 207 |
| Case of Infinite Second Moment | p. 208 |
| Saddlepoint Approximations | p. 210 |
| The t-Test and a Sequential Extension | p. 212 |
| Multivariate Extension and Hotelling's T2-Statistic | p. 213 |
| Sample Covariance Matrix and Wishart Distribution | p. 213 |
| The Multivariate t-Distribution and Hotelling's T2-Statistic | p. 213 |
| Asymptotic Theory in the Case of Non-Normal Yi | p. 215 |
| General Studentized Statistics | p. 216 |
| Martingale Central Limit Theorems and Asymptotic Normality | p. 216 |
| Non-Normal Limiting Distributions in Unit-Root Nonstationary Autoregressive Models | p. 217 |
| Studentized Statistics in Stochastic Regression Models | p. 218 |
| Supplementary Results and Problems | p. 221 |
| Self-Normalization for Approximate Pivots in Bootstrapping | p. 223 |
| Approximate Pivots and Bootstrap-t Confidence Intervals | p. 223 |
| Edgeworth Expansions and Second-Order Accuracy | p. 224 |
| Edgeworth Expansions for Smooth Functions of Sample Means | p. 224 |
| Edgeworth and Cornish-Fisher Expansions: Applications to Bootstrap-t and Percentile Intervals | p. 225 |
| Asymptotic U-Statistics and Their Bootstrap Distributions | p. 228 |
| Application of Cramer-Type Moderate Deviations | p. 232 |
| Supplementary Results and Problems | p. 233 |
| Pseudo-Maximization in Likelihood and Bayesian Inference | p. 235 |
| Generalized Likelihood Ratio Statistics | p. 235 |
| The Wilks and Wald Statistics | p. 236 |
| Score Statistics and Their Martingale Properties | p. 238 |
| Penalized Likelihood and Bayesian Inference | p. 238 |
| Schwarz's Bayesian Selection Criterion | p. 239 |
| Pseudo-Maximization and Frequentist Properties of Bayes Procedures | p. 240 |
| Supplementary Results and Problems | p. 241 |
| Sequential Analysis and Boundary Crossing Probabilities for Self-Normalized Statistics | p. 243 |
| Information Bounds and Asymptotic Optimality of Sequential GLR Tests | p. 244 |
| Likelihood Ratio Identities, the Wald-Hoeffding Lower Bounds and their Asymptotic Generalizations | p. 244 |
| Asymptotic Optimality of 2-SPRTs and Sequential GLR Tests | p. 247 |
| Asymptotic Approximations via Method of Mixtures and Geometric Integration | p. 251 |
| Boundary Crossing Probabilities for GLR Statistics via Method of Mixtures | p. 251 |
| A More General Approach Using Saddlepoint Approximations and Geometric Integration | p. 252 |
| Applications and Examples | p. 257 |
| Efficient Monte Carlo Evaluation of Boundary Crossing Probabilities | p. 260 |
| Supplementary Results and Problems | p. 262 |
| References | p. 267 |
| Index | p. 273 |
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