Moment inequalities for U-statistics. (English) Zbl 1123.60009

The paper presents moment inequalities for completely degenerate, Banach space valued U-statistics of arbitrary order. The estimates involve suprema of empirical processes which, in the real-valued case, can be replaced by simpler norms of the kernel matrix. The results generalize those of [E. Giné, R. Latala and J. Zinn, High dimensional probability II. 2nd international conference, Univ. of Washington, DC, USA, August 1–6, 1999. Boston, MA: Birkhäuser. Prog. Probab. 47, 13–38 (2000; Zbl 0969.60024)]. In the real valued case the results are used to obtain sharp estimates for moments and tails of canonical U-statistics . They are also used to obtain tail inequalities for multiple stochastic integrals of bounded deterministic functions with respect to stochastic processes with independent increments and uniformly bounded jumps.


60E15 Inequalities; stochastic orderings


Zbl 0969.60024
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[1] Adamczak, R. (2005). Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Pol. Acad. Sci. Math. 53 221–238. · Zbl 1105.60016
[2] Arcones, M. and Giné, E. (1993). On decoupling, series expansions, and tail behavior of chaos processes. J. Theoret. Probab. 6 101–122. · Zbl 0785.60023
[3] Borell, C. (1984). On a Taylor series of a Wiener polynomial. In Seminar Notes on Multiple Stochastic Integration , Polynomial Chaos and Their Integration . Case Western Reserve Univ., Cleveland. · Zbl 0573.60067
[4] Bousquet, O., Boucheron, S., Lugosi, G. and Massart, P. (2005). Moment inequalities for functions of independent random variables. Ann. Probab . 33 514–560. · Zbl 1074.60018
[5] Giné, E., Latała, R. and Zinn, J. (2000). Exponential and moment inequalities for \(U\)-statistics. In High Dimensional Probability II 13–38. Progr. Probab. 47 . Birkhäuser, Boston. · Zbl 0969.60024
[6] Giné, E., Kwapień, S., Latała, R. and Zinn, J. (2001). The LIL for canonical U-statistics of order 2. Ann. Probab. 29 520–557. · Zbl 1014.60026
[7] Houdré, C. and Reynaud-Bouret, P. (2003). Exponential inequalities, with constants, for U-statistics of order two. In Stochastic Inequalities and Applications 55–69. Prog. Probab. 56 . Birkhäuser, Basel. · Zbl 1036.60015
[8] Kwapień, S. and Woyczynski, W. (1992). Random Series and Stochastic Integrals : Single and Multiple . Birkhäuser, Boston. · Zbl 0751.60035
[9] Latała, R. (2006). Estimation of moments and tails of Gaussian chaoses. Ann. Probab. 34 2315–2331. · Zbl 1119.60015
[10] de la Peña, V. H. and Giné, E. (1999). Decoupling. From Dependence to Independence. Randomly Stopped Processes. \(U\)-Statistics and Processes. Martingales and Beyond . Springer, New York. · Zbl 0918.60021
[11] de la Peña, V. H. and Montgomery-Smith, S. (1994). Bounds on the tail probabilities of U-statistics and quadratic forms. Bull. Amer. Math. Soc. 31 223–227. · Zbl 0822.60014
[12] Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563. · Zbl 0893.60001
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