Membership of distributions of polynomials in the Nikolskii–Besov class
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 B1/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 kpolynomials 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 Bα(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.