Annals of Probability

Total variation distance between stochastic polynomials and invariance principles

Vlad Bally and Lucia Caramellino

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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 multilinear 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.

Article information

Ann. Probab., Volume 47, Number 6 (2019), 3762-3811.

Received: May 2017
Revised: June 2018
First available in Project Euclid: 2 December 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60H07: Stochastic calculus of variations and the Malliavin calculus

Stochastic polynomials invariance principles U-statistics quadratic central limit theorem abstract Malliavin calculus


Bally, Vlad; Caramellino, Lucia. Total variation distance between stochastic polynomials and invariance principles. Ann. Probab. 47 (2019), no. 6, 3762--3811. doi:10.1214/19-AOP1346.

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