U-Statistics in Banach Spaces

U-statistics are universal objects of modern probabilistic summation theory. They appear in various statistical problems and have very important applications. The mathematical nature of this class of random variables has a functional character and, therefore, leads to the investigation of probabilistic distributions in infinite-dimensional spaces. The situation when the kernel of a U-statistic takes values in a Banach space, turns out to be the most natural and interesting. In this book, the author presents in a systematic form the probabilistic theory of U-statistics with values in Banach spaces (UB-statistics), which has been developed to date. The exposition of the material in this book is based around the following topics: algebraic and martingale properties of U-statistics; inequalities; law of large numbers; the central limit theorem; weak convergence to a Gaussian chaos and multiple stochastic integrals; invariance principle and functional limit theorems; estimates of the rate of weak convergence; asymptotic expansion of distributions; large deviations; law of iterated logarithm; dependent variables; relation between Banach-valued U-statistics and functionals from permanent random measures.