Distributions of Second Order Polynomials in Gaussian Random Variables

E. D. Kosov

Matematičeskie zametki, Tome 111 (2022) no. 1, pp. 67-79.

Résumé

In this paper, we study bounds for the total variation distance between distributions of second order polynomials in normal random variables provided that they essentially depend on at least three variables. Moreover, we partially extend some recent bounds for the Kolmogorov distance between the distributions of norms of Gaussian random vectors to the case of the total variation distance.

Détail

Mots-clés : distribution of a polynomial, total variation distance
Keywords: Gaussian measure.

Bibliographie

[1] Yu. A. Davydov, G. V. Martynova, “Predelnoe povedenie raspredelenii kratnykh stokhasticheskikh integralov”, Statistika i upravlenie sluchainymi protsessami, Nauka, M., 1987, 55–57

[2] I. Nourdin, G. Poly, “Convergence in total variation on Wiener chaos”, Stoch. Process. Appl., 123:2 (2013), 651–674 | DOI | Zbl

[3] I. Nourdin, D. Nualart, G. Poly, “Absolute continuity and convergence of densities for random vectors on Wiener chaos”, Electron. J. Probab., 18 (2013), 1–19 | DOI

[4] J. C. Breton, “Convergence in variation of the joint laws of multiple Wiener–Ito integrals”, Statist. Probab. Lett., 76:17 (2006), 1904–1913 | DOI | Zbl

[5] V. I. Bogachev, E. D. Kosov, G. I. Zelenov, “Fractional smoothness of distributions of polynomials and a fractional analog of the Hardy–Landau–Littlewood inequality”, Trans. Amer. Math. Soc., 370:6 (2018), 4401–4432 | DOI | Zbl

[6] V. I. Bogachev, G. I. Zelenov, E. D. Kosov, “Prinadlezhnost raspredelenii mnogochlenov k klassam Nikolskogo–Besova”, Dokl. AN, 469:6 (2016), 651–655 | MR | Zbl

[7] E. D. Kosov, “Fractional smoothness of images of logarithmically concave measures under polynomials”, J. Math. Anal. Appl., 462:1 (2018), 390–406 | DOI | Zbl

[8] G. I. Zelenov, “On distances between distribution of polynomials”, Theory Stoch. Process, 22:2 (2017), 79–85 | Zbl

[9] V. I. Bogachev, “Raspredeleniya mnogochlenov na mnogomernykh i beskonechnomernykh prostranstvakh s merami”, UMN, 71:4 (2016), 107–154 | DOI | MR | Zbl

[10] V. I. Bogachev, “Distributions of polynomials in many variables and Nikolskii–Besov spaces”, Real Anal. Exchange, 44:1 (2019), 49–64 | DOI | Zbl

[11] V. I. Bogachev, “Mnogochleny Chebysheva–Ermita i raspredeleniya mnogochlenov ot gaussovskikh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 66:4 (2021), 693–717 | DOI

[12] E. D. Kosov, “Total variation distance estimates via $L^2$-norm for polynomials in log-concave random vectors”, Int. Math. Res. Not. (to appear)

[13] E. D. Kosov, “Otsenka mezhdu rasstoyaniyami po variatsii i v prostranstve $L^2$ dlya mnogochlenov ot logarifmicheski vognutykh sluchainykh vektorov”, Dokl. AN, 488:2 (2019), 123–125 | DOI | Zbl

[14] C. Borell, “Convex measures on locally convex spaces”, Ark. Mat., 12:1 (1974), 239–252 | DOI | Zbl

[15] V. I. Bogachev, Measure Theory, V. 1, 2, Springer, Berlin, 2007 | MR | Zbl

[16] V. Bally, L. Caramellino, “On the distances between probability density functions”, Electron. J. Probab., 19 (2014), 1–33 | DOI

[17] V. Bally, L. Caramellino, “Convergence and regularity of probability laws by using an interpolation method”, Ann. Probab., 45:2 (2017), 1110–1159 | DOI | Zbl

[18] V. Bally, L. Caramellino, “Total variation distance between stochastic polynomials and invariance principles”, Ann. Probab., 47:6 (2019), 3762–3811 | DOI | Zbl

[19] V. Bally, L. Caramellino, G. Poly, “Regularization lemmas and convergence in total variation”, Electron. J. Probab., 25 (2020), 1–20 | DOI

[20] E. D. Kosov, “On fractional regularity of distributions of functions in Gaussian random variables”, Fract. Calc. Appl. Anal., 22:5 (2019), 1249–1268 | DOI | Zbl

[21] S. Douissi, K. Es-Sebaiy, G. Kerchev, I. Nourdin, Berry–Esseen Bounds of Second Moment Estimators for Gaussian Processes Observed at High Frequency, 2021, arXiv: 2102.04810

[22] K. Es-Sebaiy, F. G. Viens, “Optimal rates for parameter estimation of stationary Gaussian processes”, Stoch. Process. Appl., 129:9 (2019), 3018–3054 | DOI | Zbl

[23] A. Olenko, V. Vaskovych, “Non-central limit theorems for functionals of random fields on hypersurfaces”, ESAIM Probab. Stat., 24 (2020), 315–340 | DOI | Zbl

[24] V. I. Bogachev, Weak Convergence of Measures, Amer. Math. Soc., Providence, RI, 2018 | MR | Zbl

[25] V. I. Bogachev, “O sekventsialnykh svoistvakh prostranstv mer”, Matem. zametki, 110:3 (2021), 459–464 | Zbl

[26] T. V. Bogachev, S. N. Popova, “Ob optimizatsii nalogovykh funktsii”, Matem. zametki, 109:2 (2021), 170–179 | Zbl

[27] V. I. Bogachev, Gaussian Measures, Math. Surveys Monogr., 62, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

[28] R. Zintout, “The total variation distance between two double Wiener–Ito integrals”, Statist. Probab. Lett., 83:10 (2013), 2160–2167 | DOI | Zbl

[29] V. I. Bogachev, E. D. Kosov, S. N. Popova, “O raspredeleniyakh odnorodnykh i vypuklykh funktsii ot gaussovskikh sluchainykh velichin”, Izv. RAN. Ser. matem., 85:5 (2021), 25–57 | DOI

[30] V. I. Bogachev, E. D. Kosov, S. N. Popova, “Plotnosti raspredelenii odnorodnykh funktsii ot gaussovskikh sluchainykh vektorov”, Dokl. AN, 495:1 (2020), 17–21 | Zbl

[31] F. Götze, A. Naumov, V. Spokoiny, V. Ulyanov, “Large ball probabilities, Gaussian comparison and anti-concentration”, Bernoulli, 25:4A (2019), 2538–2563 | MR | Zbl

[32] A. Naumov, V. Spokoiny, V. Ulyanov, “Bootstrap confidence sets for spectral projectors of sample covariance”, Probab. Theory Related Fields, 174:3 (2019), 1091–1132 | DOI | Zbl

[33] S. G. Bobkov, A. A. Naumov, V. V. Ulyanov, Two-Sided Inequalities for the Density Function's Maximum of Weighted Sum of Chi-Square Variables, 2020, arXiv: 2012.10747