The Annals of Probability

Chaos of a Markov operator and the fourth moment condition

M. Ledoux

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We analyze from the viewpoint of an abstract Markov operator recent results by Nualart and Peccati, and Nourdin and Peccati, on the fourth moment as a condition on a Wiener chaos to have a distribution close to Gaussian. In particular, we are led to introduce a notion of chaos associated to a Markov operator through its iterated gradients and present conditions on the (pure) point spectrum for a sequence of chaos eigenfunctions to converge to a Gaussian distribution. Convergence to gamma distributions may be examined similarly.

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Ann. Probab., Volume 40, Number 6 (2012), 2439-2459.

First available in Project Euclid: 26 October 2012

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Primary: 60F05: Central limit and other weak theorems 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J60: Diffusion processes [See also 58J65] 60J99: None of the above, but in this section 60H99: None of the above, but in this section

Chaos fourth moment Stein’s method Markov operator eigenfunction iterated gradient $\Gamma$-calculus


Ledoux, M. Chaos of a Markov operator and the fourth moment condition. Ann. Probab. 40 (2012), no. 6, 2439--2459. doi:10.1214/11-AOP685.

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