Abstract
The main result of this paper asserts that the distribution density of any non-constant polynomial f(ξ1,ξ2,...) of degree d in independent standard Gaussian random variables ξ1 (possibly, in infinitely many variables) always belongs to the Nikol’skii–Besov space B 1/d (R1) of fractional order 1/d (see the definition below) depending only on the degree of the polynomial. A natural analog of this assertion is obtained for the density of the joint distribution of k polynomials of degree d, also with a fractional order that is independent of the number of variables, but depends only on the degree d and the number of polynomials. We also give a new simple sufficient condition for a measure on Rk to possess a density in the Nikol’skii–Besov class B α(R)k. This result is applied for obtaining an upper bound on the total variation distance between two probability measures on Rk via the Kantorovich distance between them and a certain Nikol’skii–Besov norm of their difference. Applications are given to estimates of distributions of polynomials in Gaussian random variables.
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Published in Russian Doklady Akademii Nauk, 2016, Vol. 469, No. 6, pp. 651–655.
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Bogachev, V.I., Zelenov, G.I. & Kosov, E.D. Membership of distributions of polynomials in the Nikolskii–Besov class. Dokl. Math. 94, 453–457 (2016). https://doi.org/10.1134/S1064562416040293
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DOI: https://doi.org/10.1134/S1064562416040293