Decoupling : From Dependence to Independence

Decoupling theory provides a general framework for analyzing problems involving dependent random variables as if they were independent. In particular, decoupling is an essential component in the development of the asymptotic theory of U-- statistics and U--processes.

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.