Self-Normalized Processes [electronic resource] : Limit Theory and Statistical Applications / by Victor H. de la Peña, Tze Leung Lai, Qi-Man Shao.

Independent Random Variables -- Classical Limit Theorems, Inequalities and Other Tools -- Self-Normalized Large Deviations -- Weak Convergence of Self-Normalized Sums -- Stein's Method and Self-Normalized Berry–Esseen Inequality -- Self-Normalized Moderate Deviations and Laws of the Iterated Logarithm -- Cramér-Type Moderate Deviations for Self-Normalized Sums -- Self-Normalized Empirical Processes and U-Statistics -- Martingales and Dependent Random Vectors -- Martingale Inequalities and Related Tools -- A General Framework for Self-Normalization -- Pseudo-Maximization via Method of Mixtures -- Moment and Exponential Inequalities for Self-Normalized Processes -- Laws of the Iterated Logarithm for Self-Normalized Processes -- Multivariate Self-Normalized Processes with Matrix Normalization -- Statistical Applications -- The t-Statistic and Studentized Statistics -- Self-Normalization for Approximate Pivots in Bootstrapping -- Pseudo-Maximization in Likelihood and Bayesian Inference -- Sequential Analysis and Boundary Crossing Probabilities for Self-Normalized Statistics.

Self-normalized processes are of common occurrence in probabilistic and statistical studies. A prototypical example is Student's t-statistic introduced in 1908 by Gosset, whose portrait is on the front cover. Due to the highly non-linear nature of these processes, the theory experienced a long period of slow development. In recent years there have been a number of important advances in the theory and applications of self-normalized processes. Some of these developments are closely linked to the study of central limit theorems, which imply that self-normalized processes are approximate pivots for statistical inference. The present volume covers recent developments in the area, including self-normalized large and moderate deviations, and laws of the iterated logarithms for self-normalized martingales. This is the first book that systematically treats the theory and applications of self-normalization.