Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

Radosław Adamczak and Rafał Latała

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Abstract

We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by independent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

Résumé

Nous établissons un encadrement des moments et des queues d’un chaos polynomial d’ordre au plus trois engendré par des variables aléatoires indépendantes symétriques à queues log-concaves et pour des chaos d’ordre quelconque engendrés par des variables aléatoires indépendantes symétriques exponentielles. Ces estimations ne font intervenir que des quantités déterministes et sont optimales à des constantes près qui ne dépendent que de l’ordre du chaos.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 48, Number 4 (2012), 1103-1136.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098442

Digital Object Identifier
doi:10.1214/11-AIHP441

Mathematical Reviews number (MathSciNet)
MR3052405

Zentralblatt MATH identifier
1263.60016

Subjects
Primary: Primary 60E15 secondary 60G15

Keywords
Polynomial chaoses Tail and moment estimates Metric entropy

Citation

Adamczak, Radosław; Latała, Rafał. Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1103--1136. doi:10.1214/11-AIHP441. https://projecteuclid.org/euclid.aihp/1353098442.


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References

  • [1] R. Adamczak. Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses. Bull. Pol. Acad. Sci. Math. 53 (2005) 221–238.
  • [2] R. Adamczak. Moment inequalities for $U$-statistics. Ann. Probab. 34 (2006) 2288–2314.
  • [3] M. A. Arcones and E. Giné. On decoupling, series expansions, and tail behavior of chaos processes, J. Theoret. Probab. 6 (1993) 101–122.
  • [4] A. Bonami. Étude des coefficients de Fourier des fonctions de $L^{p}(G)$. Ann. Inst. Fourier (Grenoble) 20 (1970) 335–402.
  • [5] C. Borell. On the Taylor series of a Wiener polynomial. In Seminar Notes on Multiple Stochastic Integration, Polynomial Chaos and Their Integration. Case Western Reserve Univ., Cleveland, 1984.
  • [6] V. H. de la Peña. Decoupling and Khintchine’s inequalities for $U$-statistics. Ann. Probab. 20 (1992) 1877–1892.
  • [7] V. H. de la Peña and E. Giné. Decoupling: From Dependence to Independence. Springer, New York, 1999.
  • [8] V. H. de la Peña and S. J. Montgomery-Smith. Decoupling inequalities for the tail probabilities of multivariate $U$-statistics. Ann. Probab. 23 (1995) 806–816.
  • [9] R. M. Dudley. The sizes of compact subsets of Hilbert space and continuity of Gaussian processes. J. Funct. Anal. 1 (1967) 290–330.
  • [10] E. D. Gluskin and S. Kwapień. Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114 (1995) 303–309.
  • [11] L. Gross. Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975) 1061–1083.
  • [12] S. Kwapień. Decoupling inequalities for polynomial chaos. Ann. Probab. 15 (3) (1987) 1062–1071.
  • [13] R. Latała. Tail and moment estimates for sums of independent random vectors with logarithmically concave tails. Studia Math. 118 (1996) 301–304.
  • [14] R. Latała. Tail and moment estimates for some types of chaos. Studia Math. 135 (1999) 39–53.
  • [15] R. Latała. Estimates of moments and tails of Gaussian chaoses. Ann. Probab. 34 (2006) 2315–2331.
  • [16] R. Latała and R. Łochowski. Moment and tail estimates for multidimensional chaos generated by positive random variables with logarithmically concave tails. In Stochastic Inequalities and Applications 77–92. Progr. Probab. 56. Birkhäuser, Basel, 2003.
  • [17] M. Ledoux and M. Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Ergeb. Math. Grenzgeb. 23. Springer, Berlin, 1991.
  • [18] R. Łochowski. Moment and tail estimates for multidimensional chaoses generated by symmetric random variables with logarithmically concave tails. In Approximation and Probability 161–176. Banach Center Publ. 72. Polish Acad. Sci., Warsaw, 2006.
  • [19] E. Nelson. The free Markoff field. J. Funct. Anal. 12 (1973) 211–227.