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May 2018 Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions
Seiichiro Kusuoka, Ciprian A. Tudor
Bernoulli 24(2): 1463-1496 (May 2018). DOI: 10.3150/16-BEJ904

Abstract

We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.

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Seiichiro Kusuoka. Ciprian A. Tudor. "Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions." Bernoulli 24 (2) 1463 - 1496, May 2018. https://doi.org/10.3150/16-BEJ904

Information

Received: 1 November 2015; Revised: 1 May 2016; Published: May 2018
First available in Project Euclid: 21 September 2017

zbMATH: 06778370
MathSciNet: MR3706799
Digital Object Identifier: 10.3150/16-BEJ904

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 2 • May 2018
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