Annals of Probability

Total variation distance between stochastic polynomials and invariance principles

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
Revised: June 2018
First available in Project Euclid: 2 December 2019

https://projecteuclid.org/euclid.aop/1575277341

Digital Object Identifier
doi:10.1214/19-AOP1346

Mathematical Reviews number (MathSciNet)
MR4038042

Zentralblatt MATH identifier
07212171

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.
• [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.
• [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.
• [5] Bally, V. and Caramellino, L. (2016). An invariance principle for Stochastic series II. Non Gaussian limits. Available at arXiv:1607.04544.
• [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.
• [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.
• [11] Caravenna, F., Sun, R. and Zygouras, N. (2017). Critical polynomial chaos and marginally relevant pinning model. Private communication.
• [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.
• [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.
• [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.
• [21] Halmos, P. R. (1946). The theory of unbiased estimation. Ann. Math. Stat. 17 34–43.
• [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.
• [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.
• [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.
• [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.
• [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.