Math. Review MR2520122</a>), in the context of rates of convergence in law. Here, thanks to <i>G</i>, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for <i>Z</i> in our framework.">
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 Electronic Journal of Probability > Vol. 14 (2009) > Paper 78 open journal systems 


Density formula and concentration inequalities with Malliavin calculus

Ivan Nourdin, Université Paris 6
Frederi G Viens, Purdue University


Abstract
We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and differentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator δ (dual of the Malliavin derivative D). The formula is based on an auxilliary random variable G:=<DZ,-DL-1Z>H, where L is the generator of the so-called Ornstein-Uhlenbeck semigroup. The use of G was first discovered by Nourdin and Peccati (Probab. Theory Relat. Fields 145, 75-118, 2009, Math. Review MR2520122), in the context of rates of convergence in law. Here, thanks to G, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process. We also explain how to derive concentration inequalities for Z in our framework.


Full text: PDF

Pages: 2287-2309

Published on: October 21, 2009
Bibliography
  1. R.J. Adler. An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lecture Notes-Monograph Series, 1990. Math. Review 1088478
  2. R.J. Adler and J. Taylor. Random Fields and Geometry. Springer, 2007. Math. Review 2319516
  3. R.J. Adler; J. Taylor and K.J. Worsely. Applications of Random Fields and Geometry; Foundations and Case Studies. In preparation (2008). Math. Review number not available.
  4. J.M. Azaïs and M. Wschebor. A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail. Stoch. Proc. Appl. 118 (2008), no. 7, 1190-1218. Math. Review 2428714
  5. Ch. Borell. Tail probabilities in Gauss space. In: Vector Space Measures and Applications, Dublin, 1977. Lecture Notes in Math. 644 (1978), 73-82. Springer-Verlag. Math. Review 0502400
  6. J.-C. Breton; I. Nourdin and G. Peccati. Exact confidence intervals for the Hurst parameter of a fractional Brownian motion. Electron. J. Statist. 3 (2009), 416-425. Math. Review 2501319
  7. S. Chatterjee. Stein's method for concentration inequalities. Probab. Theory Related Fields 138 (2007), 305-321. Math. Review MR2288072
  8. S. Chatterjee. Chaos, concentration, and multiple valleys. Preprint. Math. Review number not available.
  9. L. Decreusefond and D. Nualart. Hitting times for Gaussian processes. Ann. Probab. 36, no. 1 (2008), 319-330. Math. Review 2370606
  10. C. Houdré and N. Privault. Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli 8, no. 6 (2002), 697-720. Math. Review 1962538
  11. J. Kim and D. Pollard. Cube root asymptotics. Ann. Statist. 18 (1990), 191-219. Math. Review 1041391
  12. X. Fernique. Intégrabilité des vecteurs gaussiens. C. R. Acad. Sci. Paris Sér. A-B 270, A1698-A1699. Math. Review 0266263
  13. A. Kohatsu-Higa. Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Relat. Fields 126 (2003), 421-457. Math. Review 1992500
  14. S. Kusuoka and D. Stroock. Applications of the Malliavin Calculus, Part III. J. Fac. Sci. Univ. Tokyo Sect IA Math. 34 (1987), 391-442. Math. Review 0914028
  15. I. Nourdin and G. Peccati. Stein's method on Wiener chaos. Probab. Theory Relat. Fields 145, no. 1 (2009), 75-118. Math. Review 2520122
  16. D. Nualart. The Malliavin calculus and related topics. Springer-Verlag, Berlin, second edition (2006). Math. Review 2200233
  17. E. Nualart. Exponential divergence estimates and heat kernel tail. C. R. Math. Acad. Sci. Paris 338, no. 1 (2004), 77-80. Math. Review 2038089
  18. D. Nualart and Ll. Quer-Sardanyons. Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations. Stoch. Proc. Appl., to appear. Math. Review number not available.
  19. Ch. Stein. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes - Monograph Series 7. Institute of Mathematical Statistics, Hayward, CA. Math. Review 0882007
  20. F. Viens. Stein's lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent. Stoch. Proc. Appl., to appear. Math. Review number not available.
  21. F. Viens and A. Vizcarra. Supremum Concentration Inequality and Modulus of Continuity for Sub-nth Chaos Processes. J. Funct. Anal. 248 (2007), 1-26. Math. Review 2329681
















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Electronic Journal of Probability. ISSN: 1083-6489