On Properties of Polynomials in Random Elements

Abstract

This paper deals with different properties of polynomials in random elements: bounds for characteristic functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, the characterization problem, and bounds for probabilities to hit the balls. These results cover the cases when the random elements take values in finite as well as infinite dimensional Hilbert spaces.In section 2 we give the bounds for characteristic functions gn(t, a), which improve (1) so that D increases essentially and becomes proportional to 1/m. It follows from Theorem 4 with two-sided bounds for gn(t, a) that this order of D is the right one. The improvements of (1) in section 2 are proved provided that some conditions are met concerning the type of distribution of X. A desire to remove these conditions leads to the necessity of stochastic generalization of the famous Vinogradov mean value theorem. See the stochastic generalization in section 3. In particular, if X takes values 1, 2, . . . , P , with equal probabilities, our estimate gives the same order with respect to P as Vinogradov's original result. The properties of quadratic forms are considered in sections 4 and 5. They have attracted the attention of researchers in recent years. In section 4 we give sufficient conditions at which a distribution of the finite quadratic form in independent identically distributed symmetric random variables defines a distribution of the basic random variable uniquely. The stability theorem for quadratic forms is proved as well. In section 5 the two-sided bounds are found for the density p(u, a) of |Y − a| 2 , where Y is an H-valued Gaussian random element. The bounds are precise in the sense that the ratio of upper bound to the lower equals eight for all sufficiently large values of u. Thus, the ratio does not depend on the parameters of the distribution of |Y − a| 2 . These bounds imply two-sided bounds for the probabilities P(|Y − a| > r).In this paper we use and discuss Prokhorov's results obtained jointly with various coauthors during 1995-2000 (see [6], [7], [8], [9], [10], [11]). The asymptotic behavior of quadratic and almost quadratic forms that appeared in mathematical statistics is considered in [12]. See [13] as well.

Distributions of polynomials in Gaussian random variables under structural constraints

“…This paper has been partially motivated by the papers [13] and [20], where some bounds for the characteristic functions of random variables of type (1.1) are obtained, including the following estimate (see [13,Theorem 5]):…”