Advertisement

Convergence Towards Linear Combinations of Chi-Squared Random Variables: A Malliavin-Based Approach

  • Ehsan Azmoodeh
  • Giovanni Peccati
  • Guillaume Poly
Part of the Lecture Notes in Mathematics book series (LNM, volume 2137)

Abstract

We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators \(\Gamma _{i}\) (introduced by Nourdin and Peccati, J. Appl. Funct. Anal. 258(11), 3775–3791, 2010), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly (Electron. Commun. Probab. 17(36), 1–12, 2012), concerning the limiting behavior of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.

Keywords

Malliavin Calculus Finite Linear Combination Real Separable Hilbert Space Malliavin Derivative Wiener Chaos 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

We thank P. Eichelsbacher and Ch. Thäle for discussing with us, at a preliminary stage, the results contained in [3]. EA & GP were partially supported by the Grant F1R-MTH-PUL-12PAMP (PAMPAS) from Luxembourg University.

References

  1. 1.
    H. Biermé, A. Bonami, I. Nourdin, G. Peccati, Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants. ALEA Lat. Am. J. Probab. Math. Stat. 9(2), 473–500 (2012)zbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Deya, I. Nourdin, Convergence of Wigner integrals to the tetilla law. ALEA Lat. Am. J. Probab. Math. Stat. 9, 101–127 (2012)zbMATHMathSciNetGoogle Scholar
  3. 3.
    P. Eichelsbacher, Ch. Thäle, Malliavin-Stein method for Variance-Gamma approximation on Wiener space. arXiv preprint arXiv:1409.5646, 2014Google Scholar
  4. 4.
    R.E. Gaunt, On Stein’s Method for products of normal random variables and zero bias couplings (2013), http://arxiv.org/abs/1309.4344
  5. 5.
    S. Janson, Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics, vol. 129 (Cambridge University Press, Cambridge, 1997)Google Scholar
  6. 6.
    I. Nourdin, A webpage about Stein’s method and Malliavin calculus. https://sites.google.com/site/malliavinstein/home
  7. 7.
    I. Nourdin, G. Peccati, Noncentral convergence of multiple integrals. Ann. Probab. 37(4), 1412–1426 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theory Relat. Fields. 145(1–2), 75–118 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    I. Nourdin, G. Peccati, Cumulants on the Wiener space. J. Appl. Funct. Anal. 258(11), 3775–3791 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    I. Nourdin, G. Peccati, Normal Approximations Using Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics (Cambridge University Press, Cambridge, 2012)Google Scholar
  11. 11.
    I. Nourdin, G. Poly, Convergence in law in the second Wiener/Wigner chaos. Electron. Commun. Probab. 17(36), 1–12 (2012)MathSciNetGoogle Scholar
  12. 12.
    I. Nourdin, G. Poly, Convergence in total variation on Wiener chaos. Stoch. Process. Appl. 123(2), 651–674 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Nourdin, J. Rosinski, Asymptotic independence of multiple Wiener-Ito integrals and the resulting limit laws. Ann. Probab. 42(2), 497–526 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    D. Nualart, The Malliavin Calculus and Related Topics. Probability and Its Application (Springer, Berlin, 2006)zbMATHGoogle Scholar
  15. 15.
    D. Nualart, S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stoch. Process. Appl. 118(4), 614–628 (2008)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Peccati, M.S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams (Springer, New York, 2010)Google Scholar
  18. 18.
    G. Peccati, M. Yor, Hardy’s inequality in \(L^{2}\left (\left [0,1\right ]\right )\) and principal values of Brownian local times, in Asymptotic Methods in Stochastics. Fields Institute Communications Series (American Mathematical Society, Providence, 2004), pp. 49–74Google Scholar
  19. 19.
    G. Peccati, M. Yor, Four limit theorems for quadratic functionals of Brownian motion and Brownian bridge, in Asymptotic Methods in Stochastics. Fields Institute Communication Series (American Mathematical Society, Providence, 2004), pp. 75–87Google Scholar
  20. 20.
    D. Revuz, M. Yor, Continuous Martingales and Brownian Motion (Springer, Berlin, 1999)zbMATHCrossRefGoogle Scholar
  21. 21.
    C.A. Tudor, Analysis of Variations for Self-similar Processes: A Stochastic Calculus Approach (Springer, Berlin, 2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ehsan Azmoodeh
    • 1
  • Giovanni Peccati
    • 1
  • Guillaume Poly
    • 2
  1. 1.Mathematics research unitUniversité du LuxembourgLuxembourgGermany
  2. 2.Institut de Recherche MathématiquesRENNES CedexFrance

Personalised recommendations