Arkiv för Matematik

Rademacher chaos: tail estimates versus limit theorems

Ron Blei and Svante Janson

Full-text: Open access

Abstract

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485898836

Digital Object Identifier
doi:10.1007/BF02432908

Mathematical Reviews number (MathSciNet)
MR2056543

Zentralblatt MATH identifier
1049.60007

Rights
2004 © Institut Mittag-Leffler

Citation

Blei, Ron; Janson, Svante. Rademacher chaos: tail estimates versus limit theorems. Ark. Mat. 42 (2004), no. 1, 13--29. doi:10.1007/BF02432908. https://projecteuclid.org/euclid.afm/1485898836


Export citation

References

  • 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.