The Annals of Probability

Remarks on deviation inequalities for functions of infinitely divisible random vectors

Christian Houdré

Full-text: Open access

Abstract

We obtain deviation inequalities for some classes of functions of infinitely divisible random vectors having finite exponential moments.

Article information

Source
Ann. Probab., Volume 30, Number 3 (2002), 1223-1237.

Dates
First available in Project Euclid: 20 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1029867126

Digital Object Identifier
doi:10.1214/aop/1029867126

Mathematical Reviews number (MathSciNet)
MR1920106

Zentralblatt MATH identifier
1017.60018

Subjects
Primary: 62G07: Density estimation 62C20: Minimax procedures
Secondary: 60G70: Extreme value theory; extremal processes 41A25: Rate of convergence, degree of approximation

Keywords
Infinite divisible random vectors deviation inequalities concentration of measure phenomenon Lipschitz functions

Citation

Houdré, Christian. Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 (2002), no. 3, 1223--1237. doi:10.1214/aop/1029867126. https://projecteuclid.org/euclid.aop/1029867126


Export citation

References

  • [1] BOBKOV, S. G., GÖTZE, F. and HOUDRÉ, C. (2001). On Gaussian and Bernoulli covariance representations. Bernoulli 7 439-451.
  • [2] BOBKOV, S. G. and LEDOUX, M. (1997). Poincaré inequalities and Talagrand's concentration phenomenon for the exponential measure. Probab. Theory Related Fields 107 383-400.
  • [3] BOBKOV, S. G. and LEDOUX, M. (1998). On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures. J. Funct. Anal. 156 347-365.
  • [4] CHEN, L. H. Y. (1985). Poincaré-ty pe inequalities via stochastic integrals. Z. Wahrsch. Verw. Gebiete 69 251-277.
  • [5] DE ACOSTA, A. (1980). Strong exponential inequality of sums of independent B-valued random vectors. Probab. Math. Statist. 1 133-150.
  • [6] DUDLEY, R. (1989). Real Analy sis and Probability. Wadsworth Brooks/Cole, Pacific Grove, CA.
  • [7] HOUDRÉ, C. (1998). Comparison and deviation from a representation formula. In Stochastic Processes and Related Topics: A Volume in Memory of Stamatis Cambanis, 1943-1995 (I. Karatzas, B. Rajput and M. Taqqu, eds.) 207-218. Birkhäuser, Boston.
  • [8] HOODRÉ, C. and MARCHAL, P. (2002). On the concentration of measure phenomenon for stable and related random vectors. Preprint.
  • [9] HOUDRÉ, C., PÉREZ-ABREU, V. and SURGAILIS, D. (1998). Interpolation, correlation identities and inequalities for infinitely divisible variables. J. Fourier Anal. Appl. 4 651- 668.
  • [10] HOUDRÉ, C. and PRIVAULT, N. (2001). Concentration and deviation inequalities in infinite dimensions: An approach via covariance representations. Bernoulli. To appear.
  • [11] HOUDRÉ, C. and PRIVAULT, N. (2001). A deviation inequality on Riemannian path space. Preprint.
  • [12] LEDOUX, M. (1996). Isoperimetry and Gaussian analysis. Ecole d'été de Probabilités de St. Flour 1994. Lecture Notes in Math. 1648 165-294. Springer, Berlin.
  • [13] LEDOUX, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. Séminaire de Probabilités XXXIII. Lecture Notes in Math. 1709 120-216. Springer, Berlin.
  • [14] OLVER, F. (1974). Asy mptotics and Special Functions. Academic Press, New York.
  • [15] PISIER, G. (1986). Probabilistic methods in the geometry of Banach spaces. Probability and Analy sis Lecture Notes in Math. 1206 167-241. Springer, Berlin.
  • [16] ROSI ´NSKI, J. (1995). Remarks on strong exponential integrability of vector-valued random series and triangular array s. Ann. Probab. 23 464-473.
  • [17] SATO, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press.
  • [18] TAKANO, K. (1988). On the Lévy representation of the characteristics function of the probability distribution Ce-|x| dx. Bull. Fac. Sci. Ibaraki Univ. 20 61-65.
  • [19] TALAGRAND, M. (1989). Isoperimetry and integrability of the sum of independent Banachspace valued random variables. Ann. Probab. 17 1546-1570.
  • [20] TALAGRAND, M. (1991). A new isoperimetric inequality for product measure, and the concentration of measure phenomenon. Israel Seminar (GAFA). Lecture Notes in Math. 1469 91-124. Springer, Berlin.
  • ATLANTA, GEORGIA 30332 E-MAIL: houdre@math.gatech.edu