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Full Description
01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.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.
Contents
Preface Introduction 1. BASIC DEFINITIONS One sample UB-statistics Multisample UB-statistics Von Mises' statistics Banach-valued symmetric statistics Permanent symmetric statistics Multiple stochastic integrals B-valued polynomial chaos 2. INEQUALITIES Inequalities based on the Hoeffding formula Martingale moment inequalities Maximal inequalities Contraction and symmetrization inequalities Decoupling inequalities Hypercontractive method in moment inequalities Moment inequalities in Banach spaces of type p One-sample UB-statistics Multi-sample UB-statistics Von Mises' statistics Estimates of convergence rates 4. WEAK CONVERGENCE Central limit theorem Convergence to a chaos Multi-sample UB-statistics Poisson approximation Stable approximation Approximation with increasing degrees Symmetric statistics U-statistics with varying kernels Weighted U-statistics 5. FUNCTIONAL LIMIT THEOREMS Non-degenerate kernels Degenerate kernels Weak convergence to a chaos process Weak convergence in the Poisson approximation scheme Invariance principle for symmetric statistics Functional limit theorems with varying kernels Weak convergence of U-processes 6. APPROXIMATION ESTIMATES General methods of estimation Rate of normal approximation of UR-statistics Estimates with increasing degree Nonuniform estimates Rate of chaos approximation Normal approximation of UH-statistics Multi-sample UH-statistics Estimates in central limit theorem Rate of Poisson approximation 7. ASYMPTOTIC EXPANSIONS Expansions for non-degenerate UR-statistics General method of expansions Expansions with canonical kernels Expansions with arbitrary kernels 8. LARGE DEVIATIONS Exponential inequalities Moderate deviations Power zones of normal convergence Probabilities of large deviations for UH-statistics UR-statistics UH-statistics Bounded LIL Compact LIL Functional LIL Multisample UB-statistics 10. DEPENDENT VARIABLES Symmetrically dependent random variables Weakly dependent random variables Bootstrap variables Order statistics Bibliographical supplements and comments Bibliography Index
