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Rademacher chaos: tail estimates versus limit theorems. (English) Zbl 1049.60007
Summary: We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.

60C05 Combinatorial probability
Full Text: DOI
[1] Beckner, W., Inequalities in Fourier analysis,Ann. of Math. 102 (1975), 159–182. · Zbl 0338.42017
[2] Blei, R.,Analysis in Integer and Fractional Dimensions. Cambridge Studies in Advanced Mathematics71, Cambridge Univ. Press, Cambridge, UK, 2001. · Zbl 1006.46001
[3] Blei, R. andJanson, S., Rademacher chaos: tail estimates vs limit theorems,Preprint, Institut Mittag-Leffler, Djursholm, 2001/2002.
[4] Bonami, A., Étude des coefficients de Fourier des fonctions deL p (G),Ann. Inst. Fourier (Grenoble) 20 (1970), 335–402. · Zbl 0195.42501
[5] Davie, A. M., Quotient algebras of uniform algebras,J. London Math. Soc. 7 (1973), 31–40. · Zbl 0264.46055
[6] Hanson, D. L. andWright, F. T., A bound on tail probabilities for quadratic forms in independent random variables,Ann. Math. Statist. 42 (1971), 1079–1083. · Zbl 0216.22203
[7] Hitczenko, P., Domination inequality for martingale transforms of a Rademacher sequence,Israel J. Math. 84 (1993), 161–178. · Zbl 0781.60037
[8] Janson, S., A functional limit theorem for random graphs with applications to subgraph count statistics,Random Structures Algorithms 1 (1990), 15–37. · Zbl 0708.05052
[9] Janson, S.,Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics, Memoirs Amer. Math. Soc.534, Amer. Math. Soc., Providence, RI, 1994. · Zbl 0810.05001
[10] Janson, S.,Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics129, Cambridge Univ. Press, Cambridge, 1997.
[11] Johnson, G. W. andWoodward, G. S., Onp-Sidon sets,Indiana Univ. Math. J. 24 (1974/75), 161–167.
[12] Kwapień, S. andWoyczyński, W. A.,Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, MA, 1992. · Zbl 0751.60035
[13] Latała, R., Tail and moment estimates for some types of chaos,Studia Math. 135 (1999), 39–53. · Zbl 0935.60009
[14] Littlewood, J. E., On bounded bilinear forms in an infinite number of variables,Q. J. Math. 1 (1930), 164–174. · JFM 56.0335.01
[15] McLeish, D. L., Dependent central limit theorems and invariance principles,Ann. Probab. 2 (1974), 620–628. · Zbl 0287.60025
[16] Montgomery-Smith, S. J., The distribution of Rademacher sums,Proc. Amer. Math. Soc. 109 (1990), 517–522. · Zbl 0696.60013
[17] Nelson, E., The free Markoff field,J. Funct. Anal. 12 (1973), 211–227. · Zbl 0273.60079
[18] de la Peña, V. H. andGiné, E.,Decoupling. From Dependence to Independence. Randomly Stopped Processes. U-statistics and Processes. Martingales and Beyond, Springer-Verlag, New York, 1999.
[19] Rademacher, H., Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen,Math. Ann. 87 (1922), 112–138. · JFM 48.0485.05
[20] Rubin, H. andVitale, R. A., Asymptotic distribution of symmetric statistics,Ann. Statist. 8 (1980), 165–170. · Zbl 0422.62016
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