Abstract
Nourdin and Peccati (Probab Theory Relat Fields 145(1):75–118, 2009) combined the Malliavin calculus and Stein’s method of normal approximation to associate a rate of convergence to the celebrated fourth moment theorem of Nualart and Peccati (Ann Probab 33(1):177–193, 2005). Their analysis, known as the Malliavin-Stein method nowadays, has found many applications towards stochastic geometry, statistical physics and zeros of random polynomials, to name a few. In this article, we further explore the relation between these two fields of mathematics. In particular, we construct exchangeable pairs of Brownian motions and we discover a natural link between Malliavin operators and these exchangeable pairs. By combining our findings with E. Meckes’ infinitesimal version of exchangeable pairs, we can give another proof of the quantitative fourth moment theorem. Finally, we extend our result to the multidimensional case.
Dedicated to the memory of Charles Stein, in remembrance of his beautiful mind and of his inspiring, creative, very original and deep mathematical ideas, which will, for sure, survive him for a long time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.-M. Azaïs, J.R. León, CLT for crossing of random trigonometric polynomials. Electron. J. Probab. 18(68), 1–17 (2013)
J.-M. Azaïs, F. Dalmao, J.R. León, CLT for the zeros of classical random trigonometric polynomials. Ann. Inst. H. Poincaré Probab. Stat. 52(2), 804–820 (2016)
L.H.Y. Chen, L. Goldstein, Q.M. Shao, Normal approximation by Stein’s method, in Probability and Its Applications (Springer, Berlin, 2011)
F. Dalmao, Asymptotic variance and CLT for the number of zeros of Kostlan Shub Smale random polynomials. C. R. Math. 353(12), 1141–1145 (2015)
C. Döbler, New developments in Stein’s method with applications. Ph.D. dissertation. Ruhr-Universität, Bochum, 2012. http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/DoeblerChristian/
D. Hug, G. Last, M. Schulte, Second-order properties and central limit theorems for geometric functionals of Boolean models. Ann. Appl. Probab. 26(1), 73–135 (2016)
P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in Proceeding International Symposium on Stochastic Differential Equations (Wiley, Kyoto, 1976/1978), pp. 195–263
D. Marinucci, G. Peccati, Ergodicity and Gaussianity for spherical random fields. J. Math. Phys. 51, 043301 (2010)
D. Marinucci, M. Rossi, Stein-Malliavin approximations for nonlinear functionals of random eigenfunctions on S d. J. Funct. Anal. 268(8), 2379–2420 (2015)
D. Marinucci, G. Peccati, M. Rossi, I. Wigman, Non-universality of nodal length distribution for arithmetic random waves. Geom. Funct. Anal. 26(3), 926–960 (2016)
E. Meckes, An infinitesimal version of Stein’s method of exchangeable pairs, Ph.D. dissertation. Stanford University, Stanford, 2006
E. Meckes, On Stein’s method for multivariate normal approximation, in IMS Collections, High Dimensional Probability V: The Luminy Volume, vol. 5 (2009), pp. 153–178
I. Nourdin, Malliavin-Stein approach: a webpage maintained by Ivan Nourdin. http://tinyurl.com/kvpdgcy
I. Nourdin, G. Peccati, Stein’s method on Wiener chaos. Probab. Theory Relat. Fields 145(1), 75–118 (2009)
I. Nourdin, G. Peccati, Normal approximations with Malliavin calculus: from Stein’s method to universality, in Cambridge Tracts in Mathematics, vol. 192 (Cambridge University Press, Cambridge, 2012)
I. Nourdin, J. Rosiński, Asymptotic independence of multiple Wiener-Itô integrals and the resulting limit laws. Ann. Probab. 42(2), 497–526 (2014)
I. Nourdin, G. Peccati, A. Réveillac, Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. H. Poincaré - Probab. Stat. 46(1), 45–58 (2010)
D. Nualart, The Malliavin Calculus and Related Topics, 2nd edn. Probability and Its Applications (Springer, Berlin, 2006)
D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33(1), 177–193 (2005)
G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probab. XXXVIII, 247–262 (2005)
M. Reitzner, M. Schulte, Central limit theorems for U-statistics of Poisson point processes. Ann. Probab. 41(6), 3879–3909 (2013)
A. Röllin, A note on the exchangeability condition in Stein’s method. Statist. Probab. Lett. 78, 1800–1806 (2008)
M. Schulte, A central limit theorem for the Poisson-Voronoi approximation. Adv. Appl. Math. 49(3–5), 285–306 (2012)
C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in Proceeding Sixth Berkeley Symposium on Mathematics Statistical and Probability, vol. 2 (University of California Press, California, 1972), pp. 583–602
Acknowledgements
We would like to warmly thank Christian Döbler and Giovanni Peccati, for very stimulating discussions on exchangeable pairs since the early stage of this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Nourdin, I., Zheng, G. (2019). Exchangeable Pairs on Wiener Chaos. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-030-26391-1_14
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-26390-4
Online ISBN: 978-3-030-26391-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)