# Total variation distance between stochastic polynomials and invariance principles

## TL;DR

This paper develops bounds for the total variation distance between stochastic polynomials, leading to an invariance principle that generalizes existing results and applies to U-statistics and quadratic forms.

## Contribution

It introduces a general method to estimate total variation distances between stochastic polynomials, extending previous invariance principles and CLT applications.

## Key findings

- Established bounds for total variation distance between stochastic polynomials
- Derived an invariance principle generalizing known results
- Applied results to U-statistics and quadratic forms

## Abstract

The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence one obtains an invariance principle for such polynomials. This generalizes known results concerning the total variation distance between two multiple stochastic integrals on one hand, and invariance principles in Kolmogorov distance for multi-linear stochastic polynomials on the other hand. As an application we first discuss the asymptotic behavior of U-statistics associated to polynomial kernels. Moreover we also give an example of CLT associated to quadratic forms.