Distributions of Polynomials in Gaussian Random Variables under Constraints on the Powers of Variables

E. D. Kosov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 56 (2022) no. 2, pp. 29-38.

Résumé

We study the densities of measures that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and each variable appears in the monomials of the polynomial to powers bounded by another fixed number.

Détail

Mots-clés : distribution of a polynomial, total variation distance.
Keywords: distribution density, Kantorovich distance

Bibliographie

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

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

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

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

[5] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, Nauka, M., 1996 | MR

[6] S. G. Bobkov, A. A. Naumov, V. V. Ulyanov, “Two-sided bounds for PDF's maximum of a sum of weighted chi-square variables”, Recent Developments in Stochastic Methods and Applications, ICSM-5 2020, Springer Proceedings in Mathematics and Statistics, 371, Springer, Cham., 2021, 178–189 | MR

[7] V. I. Bogachev, Gaussian Measures, Amer. Math. Soc., Providence, RI, 1998 | MR | Zbl

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

[9] V. I. Bogachev, Slabaya skhodimost mer, In-t kompyuternykh issledovanii, Izhevsk, 2016 | MR

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

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

[12] 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 | MR | Zbl

[13] V. I. Bogachev, G. I. Zelenov, “O skhodimosti po variatsii slabo skhodyaschikhsya mnogomernykh raspredelenii”, Dokl. RAN, 461:1 (2015), 14–17 | DOI | Zbl

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

[15] G. Chasapis, S. Singh, T. Tkocz, Entropies of sums of independent gamma random variables, arXiv: 2201.08284

[16] Yu. A. Davydov, G. V. Martynova, “Predelnoe povedenie raspredelenii kratnykh stokhasticheskikh integralov”, Statistika i upravlenie sluchainymi protsessami, sb. st. (Preila, 1997), Nauka, M., 1989, 55–57

[17] F. Gettse, Yu. V. Prokhorov, V. V. Ulyanov, “Otsenki dlya kharakteristicheskikh funktsii mnogochlenov ot asimptoticheski normalnykh sluchainykh velichin”, UMN, 51:2 (1996), 3–26 | DOI | MR

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

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

[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 | MR | Zbl

[21] E. D. Kosov, “Total variation distance estimates via $L^2$-norm for polynomials in log-concave random vectors”, Int. Math. Res. Not., 2021:21 (2021), 16492–16508 | DOI | MR

[22] E. D. Kosov,, “Regularity of linear and polynomial images of Skorohod differentiable measures”, Adv. Math., 397 (2022), 108193 | DOI | MR | Zbl

[23] E. D. Kosov, “Klassy Besova na konechnomernykh i beskonechnomernykh prostranstvakh”, Matem. sb., 210:5 (2019), 41–71 | DOI | MR | Zbl

[24] E. D. Kosov, “Kharakterizatsiya klassov Besova cherez novyi modul nepreryvnosti”, Dokl. RAN, 477:4 (2017), 398–401 | DOI | Zbl

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

[26] E. D. Kosov, “Raspredeleniya mnogochlenov vtoroi stepeni ot gaussovskikh sluchainykh velichin”, Matem. zametki, 111:1 (2022), 67–79 | DOI | MR | Zbl

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

[28] 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 | MR

[29] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970 | MR | Zbl

[30] V. V. Ulyanov, “O svoistvakh mnogochlenov ot sluchainykh elementov”, Teoriya veroyatn. i ee primen., 60:2 (2015), 391–402 | DOI