U-statistics in Banach spaces

Yu.V. Borovskikh

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. This book systematically presents the probabilistic theory of U-statistics with values in Banach spaces (UB-statistics), which have been developed to date. The exposition of the material 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. The book should be of interest to researchers working in probability and statistics.

「Nielsen BookData」より

[目次]

  • Part 1 Basic definitions: one sample of UB-statistics
  • multi-sample UB-statistics
  • Von Mises' statistics
  • Banach-valued symmetric statistics
  • permanent symmetric statistics
  • multiple stochastic integrals
  • B-valued polynomial chaos. Part 2 Inequalities: inqualities based on the Hoeffding formula
  • Martingale moment inequalities
  • maximal inequalities
  • contraction and symmetrization inequalities
  • de-coupling inequalities
  • hypercontractive method in moment inequalities
  • moment inequalities in Banach spaces of type r. Part 3 Law of large numbers: one-sample 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. Part 4 Functional limit theorems: non-degenerate kernels
  • degenerate kernels
  • weak convergence to a chaos process
  • weak convegence in the Poisson approximation scheme
  • invariance principle for symmetric statistics
  • functional limit theorems with varying kernels
  • weak convergence of U-processes. Paqrt 5 Approximation estimates: general methods of estimation
  • rate of normal approximation of UR-statistics
  • estimates with increasing degree
  • non-uniform estimates
  • rate of chaos approximation
  • normal approximation of UH-statistics
  • multiple-sample UH-statistics
  • estimates in central limit theorem i2r
  • rate of Poisson approximation. Part 6 Asymptotic expansions. Part 7 Large deviations. Part 8 Law of iterated logarithm. Part 9 Dependent variables. (Part contents)

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この本の情報

書名 U-statistics in Banach spaces
著作者等 Borovskikh, I︠U︡. V.
Borovskhikh Yu V.
出版元 VSP
刊行年月 1996
ページ数 xii, 420 p.
大きさ 25 cm
ISBN 9067642002
NCID BA29124392
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言語 英語
出版国 オランダ
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