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Total variation distance between stochastic polynomials and invariance principles

Vlad Bally 1, 2 Lucia Caramellino 3
2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
Abstract : The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence, one obtains an in-variance 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 multilin-ear 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.
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Vlad Bally, Lucia Caramellino. Total variation distance between stochastic polynomials and invariance principles. Annals of Probability, Institute of Mathematical Statistics, 2019, 47, pp.3762 - 3811. ⟨10.1214/19-AOP1346⟩. ⟨hal-02429560⟩

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