Annals of Probability
- Ann. Probab.
- Volume 47, Number 6 (2019), 3762-3811.
Total variation distance between stochastic polynomials and invariance principles
Vlad Bally and Lucia Caramellino
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Abstract
The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence, one obtains an invariance principle for such polynomials. This generalizes known results concerning the total variation distance between two multiple stochastic integrals on one hand, and invariance principles in Kolmogorov distance for multilinear stochastic polynomials on the other hand. As an application, we first discuss the asymptotic behavior of U-statistics associated to polynomial kernels. Moreover, we also give an example of CLT associated to quadratic forms.
Article information
Source
Ann. Probab., Volume 47, Number 6 (2019), 3762-3811.
Dates
Received: May 2017
Revised: June 2018
First available in Project Euclid: 2 December 2019
Permanent link to this document
https://projecteuclid.org/euclid.aop/1575277341
Digital Object Identifier
doi:10.1214/19-AOP1346
Mathematical Reviews number (MathSciNet)
MR4038042
Zentralblatt MATH identifier
07212171
Subjects
Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Keywords
Stochastic polynomials invariance principles U-statistics quadratic central limit theorem abstract Malliavin calculus
Citation
Bally, Vlad; Caramellino, Lucia. Total variation distance between stochastic polynomials and invariance principles. Ann. Probab. 47 (2019), no. 6, 3762--3811. doi:10.1214/19-AOP1346. https://projecteuclid.org/euclid.aop/1575277341
References
- [1] Azmoodeh, E., Peccati, G. and Poly, G. (2016). The law of iterated logarithm for subordinated Gaussian sequences: Uniform Wasserstein bounds. ALEA Lat. Am. J. Probab. Math. Stat. 13 659–686.Zentralblatt MATH: 1346.60030
- [2] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 348. Springer, Cham.Zentralblatt MATH: 1376.60002
- [3] Bally, V. and Caramellino, L. (2014). On the distances between probability density functions. Electron. J. Probab. 19 no. 110, 33.
- [4] Bally, V. and Caramellino, L. (2016). Asymptotic development for the CLT in total variation distance. Bernoulli 22 2442–2485.Zentralblatt MATH: 1346.60016
Digital Object Identifier: doi:10.3150/15-BEJ734
Project Euclid: euclid.bj/1462297686 - [5] Bally, V. and Caramellino, L. (2016). An invariance principle for Stochastic series II. Non Gaussian limits. Available at arXiv:1607.04544.arXiv: 1607.04544
Zentralblatt MATH: 1371.60096
Digital Object Identifier: doi:10.1007/978-3-319-27128-6 - [6] Bally, V., Caramellino, L. and Poly, G. (2018). Convergence in distribution norms in the CLT for non identical distributed random variables. Electron. J. Probab. 23 Paper No. 45, 51.
- [7] Bally, V. and Rey, C. (2016). Approximation of Markov semigroups in total variation distance. Electron. J. Probab. 21 Paper No. 12, 44.
- [8] Bentkus, V. (2004). On Hoeffding’s inequalities. Ann. Probab. 32 1650–1673.Zentralblatt MATH: 1062.60011
Digital Object Identifier: doi:10.1214/009117904000000360
Project Euclid: euclid.aop/1084884866 - [9] Bogachev, V. I., Kosov, E. D. and Zelenov, G. I. (2018). Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality. Trans. Amer. Math. Soc. 370 4401–4432.
- [10] Caravenna, F., Sun, R. and Zygouras, N. (2017). Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 3050–3112.Zentralblatt MATH: 1387.82032
Digital Object Identifier: doi:10.1214/17-AAP1276
Project Euclid: euclid.aoap/1509696041 - [11] Caravenna, F., Sun, R. and Zygouras, N. (2017). Critical polynomial chaos and marginally relevant pinning model. Private communication.Zentralblatt MATH: 1387.82032
Digital Object Identifier: doi:10.1214/17-AAP1276
Project Euclid: euclid.aoap/1509696041 - [12] Carbery, A. and Wright, J. (2001). Distributional and $L^{q}$ norm inequalities for polynomials over convex bodies in $\mathbb{R}^{n}$. Math. Res. Lett. 8 233–248.
- [13] Davydov, Y. (2017). On distance in total variation between image measures. Statist. Probab. Lett. 129 393–400.
- [14] Davydov, Y. A. and Martynova, G. V. (1989). Limit behavior of distributions of multiple stochastic integrals. In Statistics and Control of Random Processes (Russian) (Preila, 1987) 55–57. “Nauka”, Moscow.
- [15] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277.
- [16] de Jong, P. (1990). A central limit theorem for generalized multilinear forms. J. Multivariate Anal. 34 275–289.Mathematical Reviews (MathSciNet): MR1073110
Zentralblatt MATH: 0709.60019
Digital Object Identifier: doi:10.1016/0047-259X(90)90040-O - [17] Döbler, C. and Peccati, G. (2017). Quantitative de Jong theorems in any dimension. Electron. J. Probab. 22 Paper No. 2, 35.
- [18] Döbler, C. and Peccati, G. (2018). The gamma Stein equation and noncentral de Jong theorems. Bernoulli 24 3384–3421.Zentralblatt MATH: 1407.60034
Digital Object Identifier: doi:10.3150/17-BEJ963
Project Euclid: euclid.bj/1524038757 - [19] Gamkrelidze, N. G. and Rotar’, V. I. (1977). The rate of convergence in a limit theorem for quadratic forms. Teor. Veroyatn. Primen. 22 404–407.
- [20] Götze, F. and Tikhomirov, A. N. (1999). Asymptotic distribution of quadratic forms. Ann. Probab. 27 1072–1098.Zentralblatt MATH: 0941.60049
Digital Object Identifier: doi:10.1214/aop/1022677395
Project Euclid: euclid.aop/1022677395 - [21] Halmos, P. R. (1946). The theory of unbiased estimation. Ann. Math. Stat. 17 34–43.Zentralblatt MATH: 0063.01891
Digital Object Identifier: doi:10.1214/aoms/1177731020
Project Euclid: euclid.aoms/1177731020 - [22] Hoeffding, W. (1961). The Strong Law of Large Numbers for U-Statistics. Institute of Statistics, Mimeo-Series No. 302. Univ. North Carolina, Chapel HIll, NC.
- [23] Lee, A. J. (1990). $U$-Statistics: Theory and Practice. Statistics: Textbooks and Monographs 110. Dekker, New York.
- [24] Löcherbach, E. and Loukianova, D. (2008). On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stochastic Process. Appl. 118 1301–1321.
- [25] Mossel, E., O’Donnell, R. and Oleszkiewicz, K. (2010). Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. (2) 171 295–341.
- [26] Noreddine, S. and Nourdin, I. (2011). On the Gaussian approximation of vector-valued multiple integrals. J. Multivariate Anal. 102 1008–1017.
- [27] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
- [28] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge Univ. Press, Cambridge.Zentralblatt MATH: 1266.60001
- [29] Nourdin, I., Peccati, G., Poly, G. and Simone, R. (2016). Classical and free fourth moment theorems: Universality and thresholds. J. Theoret. Probab. 29 653–680.
- [30] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947–1985.Zentralblatt MATH: 1246.60039
Digital Object Identifier: doi:10.1214/10-AOP531
Project Euclid: euclid.aop/1282053777 - [31] Nourdin, I., Peccati, G. and Réveillac, A. (2010). Multivariate normal approximation using Stein’s method and Malliavin calculus. Ann. Inst. Henri Poincaré Probab. Stat. 46 45–58.
- [32] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651–674.
- [33] Nourdin, I. and Poly, G. (2015). An invariance principle under the total variation distance. Stochastic Process. Appl. 125 2190–2205.
- [34] Nualart, D. and Ortiz-Latorre, S. (2008). Central limit theorems for multiple stochastic integrals and Malliavin calculus. Stochastic Process. Appl. 118 614–628.
- [35] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.Zentralblatt MATH: 1097.60007
Digital Object Identifier: doi:10.1214/009117904000000621
Project Euclid: euclid.aop/1108141724 - [36] Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 43 309–318.
- [37] Peccati, G. and Tudor, C. A. (2005). Gaussian limits for vector-valued multiple stochastic integrals. In Séminaire de Probabilités XXXVIII. Lecture Notes in Math. 1857 247–262. Springer, Berlin.
- [38] Poly, G. (2012). Dirichlet forms and applications to the ergodic theory of Markov chains. Ph.D. thesis, https://tel.archives-ouvertes.fr/tel-00690724.
- [39] Prokhorov, Y. V. (1952). A local theorem for densities. Dokl. Akad. Nauk SSSR 83 797–800.Zentralblatt MATH: 0046.35301
- [40] Rotar’, V. I. (1975). Limit theorems for multilinear forms and quasipolynomial functions. Teor. Veroyatn. Primen. 20 527–546.
- [41] Rotar’, V. I. and Shervashidze, T. L. (1985). Some estimates for distributions of quadratic forms. Teor. Veroyatn. Primen. 30 549–554.Zentralblatt MATH: 0578.60020

- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- On the distances between probability density functions
Bally, Vlad and Caramellino, Lucia, Electronic Journal of Probability, 2014 - Moment inequalities for U-statistics
Adamczak, Radosław, Annals of Probability, 2006 - Chapter V. Theory of a Single Linear Transformation
Anthony W. Knapp, Basic Algebra, Digital Second Edition (East Setauket, NY: Anthony W. Knapp, 2016), 2016
- On the distances between probability density functions
Bally, Vlad and Caramellino, Lucia, Electronic Journal of Probability, 2014 - Moment inequalities for U-statistics
Adamczak, Radosław, Annals of Probability, 2006 - Chapter V. Theory of a Single Linear Transformation
Anthony W. Knapp, Basic Algebra, Digital Second Edition (East Setauket, NY: Anthony W. Knapp, 2016), 2016 - Quadratic transformations for orthogonal polynomials in one and two variables
Koornwinder, Tom H., , 2018 - Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension
Da Prato, Giuseppe and Lunardi, Alessandra, Annals of Probability, 2014 - Random surface growth and Karlin-McGregor polynomials
Assiotis, Theodoros, Electronic Journal of Probability, 2018 - On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions
Kojima, Hisashi and Tokuno, Yasushi, Tohoku Mathematical Journal, 2004 - Hoffmann's conjecture for totally singular forms of prime degree
Scully, Stephen, Algebra & Number Theory, 2016 - ANOVA for diffusions and Itô processes
Mykland, Per Aslak and Zhang, Lan, Annals of Statistics, 2006 - Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations
Hairer, Martin and Mattingly, Jonathan C., Annals of Probability, 2008