Decoupling

This monograph presents important recent results in the areas of pure and applied probability. The authors are recognized experts in this area.

A friendly and systematic introduction to the theory and applications of decoupling Special emphasis is given to the comparison and interplay between martingale and decoupling theories Applications are emphasized and include results with biostatistical implications Authors are recognized experts in this area

Zusammenfassung
Randomly Stopped Processes U-Statistics and Processes Martingales and Beyond

Inhalt
1 Sums of Independent Random Variables.- 1.1 Lévy-Type Maximal Inequalities.- 1.2 Hoffmann-J?rgensen Type Inequalities.- 1.3 The KhinchinKahane Inequalities.- 1.4 Moment Bounds.- 1.5 Estimates with Sharp Constants for the La-Norms of Sums of Independent Random Variables: The L-Function.- 1.6 References for Chapter 1.- 2 Randomly Stopped Processes With Independent Increments.- 2.1 Wald's Equations.- 2.2 Good-Lambda Inequalities.- 2.3 Randomly Stopped Sums of Independent Banach-Valued Variables.- 2.4 Proof of the Lower Bound of Theorem 2.3.1.- 2.5 Continuous Time Processes.- 2.6 BurkholderGundy Type Inequalities in Banach Spaces.- 2.7 From Boundary Crossing of Nonrandom Functions to First Passage Times of Processes with Independent Increments.- 2.8 References for Chapter 2.- 3 Decoupling of U-Statistics and U-Processes.- 3.1 Decoupling of U-Processes: Convex Functions.- 3.2 Hypercontractivity of Rademacher Chaos Variables.- 3.3 Minorization of Tail Probabilities: The PaleyZygmund Argument and a Conditional Jensen's Inequality.- 3.4 Decoupling of U-processes: Tail Probabilities.- 3.5 Randomization136.- 3.6 References for Chapter 3.- 4 Limit Theorems for U-Statistics.- 4.1 Some Inequalities; the Law of Large Numbers.- 4.2 Gaussian Chaos and the Central Limit Theorem for Canonical U-Statistics.- 4.3 The Law of the Iterated Logarithm for Canonical U-Statistics.- 4.4 References for Chapter 4.- 5 Limit Theorems for U-Processes.- 5.1 Some Background on Asymptotics of Processes, Metric Entropy, and Vapnik?ervonenkis Classes of Functions: Maximal Inequalities.- 5.2 The Law of Large Numbers for U-Processes.- 5.3 The Central Limit Theorem for U-Processes.- 5.4 The Law of the Iterated Logarithm for Canonical U-Processes.- 5.5 Statistical Applications.- 5.6References for Chapter 5.- 6 General Decoupling Inequalities for Tangent Sequences.- 6.1 Some Definitions and Examples.- 6.2 Exponential Decoupling Inequalities for Sums.- 6.3 Tail Probability andLpInequalities for Tangent Sequences I.- 6.4 Tail Probability and Moment Inequalities for Tangent Sequences II: Good-Lambda Inequalities.- 6.5 Differential Subordination and Applications.- 6.6 Decoupling Inequalities Compared to Martingale Inequalities.- 6.7 References for Chapter 6323.- 7 Conditionally Independent Sequences.- 7.1 The Principle of Conditioning and Related Results.- 7.2 Analysis of a Sequence of Two-by-Two Tables.- 7.3 SharpLpComparison of Sums of Arbitrarily Dependent Variables to Sums of CI Variables.- 7.4 References for Chapter 7.- 8 Further Applications of Decoupling.- 8.1 Randomly Stopped Canonical U-Statistics.- 8.2 A General Class of Exponential Inequalities for Martingales and Ratios.- 8.3 References for Chapter 8.- References.