On Properties of Polynomials in Random Elements

TL;DR

This paper investigates various properties of polynomials in random elements, providing bounds and generalizations applicable to finite and infinite-dimensional spaces, advancing understanding in stochastic analysis.

Contribution

It introduces new bounds and a stochastic generalization of the Vinogradov mean value theorem for polynomials in random elements across different spaces.

Findings

Bounds for characteristic functionals of polynomials

Stochastic Vinogradov mean value theorem generalization

Bounds for hitting probabilities in Hilbert spaces

Abstract

The paper deals with different properties of polynomials in random elements: bounds for characteristics functionals of polynomials, stochastic generalization of the Vinogradov mean value theorem, characterization problem, bounds for probabilities to hit the balls. These results cover the cases where the random elements take values in finite as well as infinite dimensional Hilbert spaces.