Decoupling: From Dependence to Independence

ISBN-13: 9781461268086 / Angielski / Miękka / 2012 / 392 str.

Victor De La Pena; Evarist Gine; Victor de La Pena

Opis

A friendly and systematic introduction to the theory and applications. The book begins with the sums of independent random variables and vectors, with maximal inequalities and sharp estimates on moments, which are later used to develop and interpret decoupling inequalities. Decoupling is first introduced as it applies to randomly stopped processes and unbiased estimation. The authors then proceed with the theory of decoupling in full generality, paying special attention to comparison and interplay between martingale and decoupling theory, and to applications. These include limit theorems, moment and exponential inequalities for martingales and more general dependence structures, biostatistical implications, and moment convergence in Anscombe's theorem and Wald's equation for U--statistics. Addressed to researchers in probability and statistics and to graduates, the expositon is at the level of a second graduate probability course, with a good portion of the material fit for use in a first year course.

Recenzje

From a review:

MATHEMATICAL REVIEWS

"The book is written in an excellent way. The exposition is clear and effective. The results are well motivated."

Spis treści

1 Sums of Independent Random Variables.- 1.1 Lévy-Type Maximal Inequalities.- 1.2 Hoffmann-J?rgensen Type Inequalities.- 1.3 The Khinchin—Kahane Inequalities.- 1.4 Moment Bounds.- 1.4.1 Maxima.- 1.4.2 Estimating La-Norms in Hilbert Space.- 1.4.3 K-Function Bounds.- 1.4.4 A General Decoupling for Sums of Arbitrary Positive Random Variables and Martingales.- 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 Burkholder—Gundy 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 Paley—Zygmund Argument and a Conditional Jensen’s Inequality.- 3.4 Decoupling of U-processes: Tail Probabilities.- 3.5 Randomization136.- 3.5.1 Moment Inequalities for Randomized U-Statistics139.- 3.5.2 Randomization of Tail Probabilities for U-Statistics and Processes.- 3.6 References for Chapter 3.- 4 Limit Theorems for U-Statistics.- 4.1 Some Inequalities; the Law of Large Numbers.- 4.1.1 Hoffmann-J?rgensen’s Inequality for U-Processes.- 4.1.2 An Application to the Law of Large Numbers.- 4.1.3 Exponential Inequalities for Canonical U-Statistics.- 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.- 5.1.1 Convergence in Law of Sample Bounded Processes.- 5.1.2 Maximal Inequalities Based on Metric Entropy.- 5.1.3 Vapnik—?ervonenkis Classes of Functions.- 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.- 5.4.1 The Bounded LIL.- 5.4.2 The Compact LIL.- 5.5 Statistical Applications.- 5.5.1 The Law of Large Numbers for the Simplicial Median.- 5.5.2 The Central Limit Theorem for the Simplicial Median.- 5.5.3 Truncated Data.- 5.6 References 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.- 8.1.1 Wald’s Equation for Canonical U-Statistics.- 8.1.2 Moment Bounds for Regular and Randomly StoppedU-Statistics.- 8.1.3 Moment Convergence in Anscombe’s Theorem forU-Statistics.- 8.2 A General Class of Exponential Inequalities for Martingales and Ratios.- 8.3 References for Chapter 8.- References.