Arkiv för Matematik

Rademacher chaos: tail estimates versus limit theorems

Ron Blei and Svante Janson

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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.

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Ark. Mat., Volume 42, Number 1 (2004), 13-29.

Received: 30 September 2002
Revised: 26 May 2003
First available in Project Euclid: 31 January 2017

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2004 © Institut Mittag-Leffler


Blei, Ron; Janson, Svante. Rademacher chaos: tail estimates versus limit theorems. Ark. Mat. 42 (2004), no. 1, 13--29. doi:10.1007/BF02432908.

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  • Beckner, W., Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159–182.
  • Blei, R., Analysis in Integer and Fractional Dimensions. Cambridge Studies in Advanced Mathematics 71, Cambridge Univ. Press, Cambridge, UK, 2001.
  • Blei, R. and Janson, S., Rademacher chaos: tail estimates vs limit theorems, Preprint, Institut Mittag-Leffler, Djursholm, 2001/2002.
  • Bonami, A., Étude des coefficients de Fourier des fonctions de Lp(G), Ann. Inst. Fourier (Grenoble) 20 (1970), 335–402.
  • Davie, A. M., Quotient algebras of uniform algebras, J. London Math. Soc. 7 (1973), 31–40.
  • Hanson, D. L. and Wright, F. T., A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), 1079–1083.
  • Hitczenko, P., Domination inequality for martingale transforms of a Rademacher sequence, Israel J. Math. 84 (1993), 161–178.
  • Janson, S., A functional limit theorem for random graphs with applications to subgraph count statistics, Random Structures Algorithms 1 (1990), 15–37.
  • Janson, S., Orthogonal Decompositions and Functional Limit Theorems for Random Graph Statistics, Memoirs Amer. Math. Soc. 534, Amer. Math. Soc., Providence, RI, 1994.
  • Janson, S., Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics 129, Cambridge Univ. Press, Cambridge, 1997.
  • Johnson, G. W. and Woodward, G. S., On p-Sidon sets, Indiana Univ. Math. J. 24 (1974/75), 161–167.
  • Kwapień, S. and Woyczyński, W. A., Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston, MA, 1992.
  • Latała, R., Tail and moment estimates for some types of chaos, Studia Math. 135 (1999), 39–53.
  • Littlewood, J. E., On bounded bilinear forms in an infinite number of variables, Q. J. Math. 1 (1930), 164–174.
  • McLeish, D. L., Dependent central limit theorems and invariance principles, Ann. Probab. 2 (1974), 620–628.
  • Montgomery-Smith, S. J., The distribution of Rademacher sums, Proc. Amer. Math. Soc. 109 (1990), 517–522.
  • Nelson, E., The free Markoff field, J. Funct. Anal. 12 (1973), 211–227.
  • de la Peña, V. H. and Giné, E., Decoupling. From Dependence to Independence. Randomly Stopped Processes. U-statistics and Processes. Martingales and Beyond, Springer-Verlag, New York, 1999.
  • Rademacher, H., Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen, Math. Ann. 87 (1922), 112–138.
  • Rubin, H. and Vitale, R. A., Asymptotic distribution of symmetric statistics, Ann. Statist. 8 (1980), 165–170.